%%%% How to reproduce a hard copy of this paper: %%%% %%%% This paper is in PostScript format. %%%% Remove lines above the "cut here" line; save this file %%%% as nonlocal.ps, and use "lpr nonlocal.ps" %%%% (or some command like it) to print. %%%% %%%%%%%%%%%%%%%%%%% cut here %%%%%%%%%%%%%%%% %!PS-Adobe-2.0 %%Creator: dvipsk 5.55a Copyright 1986, 1994 Radical Eye Software %%Title: nonlocal.dvi %%Pages: 25 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips nonlocal -o %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1995.10.18:1947 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse 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b(equations)e(are)f(appro)n(ximate)g(reductions)h(of) g(complicated)h(ph)n(ysical)e(sys-)523 896 y(tems)22 b(on)f(extended)h(domains)f(near)f(bifurcation)i(p)r(oin)n(ts)f(\(see)g (V)-7 b(ega)21 b([1993],)g(New)n(ell)h(et)g(al.)523 995 y([1993])33 b(and)h(references)f(therein\).)58 b(Amplitude)36 b(equations)d(usually)h(ha)n(v)n(e)g(\\univ)n(ersal")523 1095 y(forms)24 b(and)g(therefore,)g(one)g(amplitude)h(equation)e(ma)n (y)h(capture)g(imp)r(ortan)n(t)g(features)f(of)523 1195 y(v)-5 b(arious)24 b(ph)n(ysical)g(systems.)35 b(The)25 b(deriv)-5 b(ation)24 b(of)h(suc)n(h)g(amplitude)g(equations)f(is)h (usually)523 1294 y(based)k(on)g(a)f(formal)h(asymptotic)f(expansion)h (with)g(resp)r(ect)g(to)g(a)g(small)g(parameter.)40 b(A)523 1394 y(rigorous)24 b(justi\014cation)j(for)f(suc)n(h)g(deriv)-5 b(ation)27 b(is)f(presen)n(ted)g(in)h(T)-7 b(ak\023)-42 b(a)n(\024)j(c)24 b(et)j(al.)37 b([1995])24 b(\(see)523 1494 y(also)j(references)f(therein\).)648 1593 y(The)18 b(nonlinear)f(Sc)n(hr\177)-42 b(odinger)17 b(equation)h(and)g(the)h (complex)f(Ginzburg-Landau)f(equa-)523 1693 y(tion)31 b(are)g(examples)f(of)i(amplitude)g(equations,)f(whic)n(h)g(are)g(in)g (lo)r(cal)g(forms.)47 b(The)32 b(non-)523 1792 y(linear)24 b(Sc)n(hr\177)-42 b(odinger)22 b(equation)i(comes)g(from)g(conserv)-5 b(ativ)n(e)22 b(ph)n(ysical)i(systems)f(while)i(the)523 1892 y(Ginzburg-Landau)19 b(equation)h(comes)g(from)g(dissipativ)n(e)g (ones.)34 b(In)21 b(some)f(situations)g(non-)523 1992 y(lo)r(cal)j(in)n(tegral)f(terms)i(ma)n(y)f(app)r(ear)f(in)i(the)g (amplitude)g(equations.)35 b(F)-7 b(or)23 b(example,)h(if)g(the)523 2091 y(original)30 b(ph)n(ysical)h(system)g(has)g(some)g(kind)g(of)h (symmetries,)g(in)n(tegral)e(terms)h(ma)n(y)g(ap-)523 2191 y(p)r(ear)26 b(in)g(the)h(deriv)n(ed)e(amplitude)i(equation)f (\(Matk)n(o)n(wsky)e(and)i(V)-7 b(olp)r(ert)27 b([1992]\).)35 b(Some)523 2291 y(deriv)-5 b(ation)34 b(metho)r(ds)g(ma)n(y)g(also)f (in)n(tro)r(duce)h(in)n(tegral)f(terms)h(\(Mielk)n(e)g([1992]\).)56 b(When)523 2390 y(the)22 b(original)e(ph)n(ysical)h(systems)g(con)n (tain)f(nonlo)r(cal)h(terms,)i(the)e(reduced)h(or)e(deriv)n(ed)h(am-) 523 2490 y(plitude)k(equations)f(ma)n(y)g(con)n(tain)g(similar)g(nonlo) r(cal)g(terms)g(as)g(in)h(Elmer)f([1988].)34 b(Elmer)523 2589 y([1988])28 b(deriv)n(ed)h(the)h(follo)n(wing)f(amplitude)h (equation)g(for)f(a)g(class)g(of)h(ph)n(ysical)f(systems)523 2689 y(including)f(a)f(ferromagnetic)f(system:)632 2923 y Fj(u)680 2935 y Fi(t)732 2923 y Fs(=)d Fj(a)864 2935 y Fr(0)901 2923 y Fj(u)18 b Fs(+)g(\()p Fj(a)1126 2935 y Fr(1)1182 2923 y Fs(+)g Fj(ib)1330 2935 y Fr(1)1366 2923 y Fs(\))p Fj(u)1446 2935 y Fi(xx)1544 2923 y Fs(+)g(\()p Fj(a)1703 2935 y Fr(2)1759 2923 y Fs(+)g Fj(ib)1907 2935 y Fr(2)1943 2923 y Fs(\))p Fh(j)p Fj(u)p Fh(j)2069 2889 y Fr(2)2106 2923 y Fj(u)g Fs(+)g(\()p Fj(a)2331 2935 y Fr(3)2387 2923 y Fs(+)g Fj(ib)2535 2935 y Fr(3)2572 2923 y Fs(\))p Fj(u)2662 2867 y Fs(1)p 2662 2904 42 4 v 2670 2980 a Fj(l)2727 2810 y Fg(Z)2810 2831 y Fi(l)2773 2999 y Fr(0)2849 2923 y Fh(j)p Fj(u)p Fh(j)2943 2889 y Fr(2)2980 2923 y Fj(dx;)124 b Fs(\(1.1\))523 3142 y(sub)5 b(ject)28 b(to)f(the)h(p)r(erio)r(dic)g(b)r(oundary)f(condition)1591 3318 y Fj(u)p Fs(\()p Fj(x)19 b Fs(+)f Fj(l)r(;)c(t)p Fs(\))22 b(=)h Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))p Fj(;)912 b Fs(\(1.2\))523 3495 y(and)27 b(supplemen)n(ted)h(with)h (initial)e(condition)1664 3671 y Fj(u)p Fs(\()p Fj(x;)14 b Fs(0\))23 b(=)g Fj(u)2061 3683 y Fr(0)2097 3671 y Fs(\()p Fj(x)p Fs(\))p Fj(;)986 b Fs(\(1.3\))523 3848 y(for)22 b(some)g(\014xed)h Fj(l)h(>)f Fs(0,)g(and)f(for)g(ev)n(ery)g Fj(x)h Fh(2)g Fj(I)-21 b(R)q Fs(,)24 b Fj(t)f(>)g Fs(0.)35 b(The)22 b(function)h Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))23 b(is)g(complex)523 3947 y(and)34 b(the)g(co)r(e\016cien)n(ts)g Fj(a)1311 3959 y Fi(j)1346 3947 y Fs('s,)h Fj(b)1496 3959 y Fi(j)1531 3947 y Fs('s)f(are)f(real.)55 b(When)35 b Fj(a)2272 3959 y Fr(3)2343 3947 y Fs(=)e Fj(b)2477 3959 y Fr(3)2548 3947 y Fs(=)g(0)h(the)g(equation)g(\(1.1\))523 4047 y(reduces)27 b(to)g(the)h(cubic)g(complex)f(Ginzburg-Landau)f (equation,)1183 4224 y Fj(u)1231 4236 y Fi(t)1283 4224 y Fs(=)c Fj(a)1414 4236 y Fr(0)1451 4224 y Fj(u)c Fs(+)g(\()p Fj(a)1676 4236 y Fr(1)1732 4224 y Fs(+)g Fj(ib)1880 4236 y Fr(1)1917 4224 y Fs(\))p Fj(u)1997 4236 y Fi(xx)2094 4224 y Fs(+)g(\()p Fj(a)2253 4236 y Fr(2)2309 4224 y Fs(+)g Fj(ib)2457 4236 y Fr(2)2494 4224 y Fs(\))p Fh(j)p Fj(u)p Fh(j)2620 4189 y Fr(2)2657 4224 y Fj(u:)489 b Fs(\(1.4\))648 4400 y(Other)34 b(nonlo)r(cal)g(in)n(tegro-di\013eren)n (tial)f(amplitude)i(equations)f(also)g(app)r(ear)g(in)h(the)523 4500 y(mo)r(deling)h(of)g(in)n(terfacial)g(phenomena)f(\(P)n (apageorgiou)e(et)k(al.)62 b([1990]\),)37 b(fron)n(tal)e(phe-)523 4599 y(nomena)40 b(\(Wilder)g(et)g(al.)74 b([1994])39 b(and)h(Metzener)g(et)g(al.)74 b([1994]\),)42 b(and)e(bifurcation)523 4699 y(phenomena)27 b(\(Or-Guil)g(et)h(al.)37 b([1994]\).)648 4799 y(A)i(global)f(attractor)f(for)h(an)h(ev)n(olutionary)e(partial)h (di\013eren)n(tial)h(equation)f(is)h(the)523 4898 y(maximal)30 b(b)r(ounded)g(in)n(v)-5 b(arian)n(t)29 b(set,)i(whic)n(h)f(attracts)g (all)g(solution)f(orbits)h(in)g(the)h(phase)1934 5147 y(2)p eop %%Page: 3 3 3 2 bop 523 614 a Fs(space)23 b(or)g(functional)h(space.)35 b(The)24 b(global)e(attractor)g(needs)i(not)g(to)f(b)r(e)i(a)e(smo)r (oth)g(subset)523 714 y(of)28 b(the)h(phase)f(space)g(and)g(it)h(ma)n (y)f(b)r(e)h(v)n(ery)e(complicated)h(or)g(ev)n(en)g(fractal.)39 b(An)29 b(inertial)523 814 y(manifold)22 b(for)g(an)g(ev)n(olutionary)e (equation)i(is,)h(on)f(the)h(other)f(hand,)h(a)f(\014nite)h (dimensional)523 913 y(Lipsc)n(hitz)35 b(manifold,)i(whic)n(h)e(is)f(p) r(ositiv)n(ely)h(in)n(v)-5 b(arian)n(t)34 b(under)h(the)g(solution)f (\015o)n(w,)j(and)523 1013 y(whic)n(h)e(exp)r(onen)n(tially)g(attracts) g(all)g(b)r(ounded)h(sets)f(in)h(the)f(phase)g(space.)60 b(See)36 b(F)-7 b(oias-)523 1112 y(Sell-T)g(emam)29 b([1988],)e(F)-7 b(oias,)28 b(Sell)h(and)g(Titi)g([1989],)e(T)-7 b(emam)29 b([1988],)f(Hale)g([1988],)g(and)523 1212 y(Constan)n(tin)e(et)g(al.)36 b([1989])25 b(for)h(more)f(details.)36 b(As)27 b(a)e(result)h(it)h (necessarily)e(con)n(tains)g(the)523 1312 y(global)d(attractor.)33 b(The)23 b(\015o)n(w)f(restricted)g(to)h(an)f(inertial)h(manifold)f(is) h(equiv)-5 b(alen)n(t)22 b(to)h(that)523 1411 y(of)e(a)g(\014nite)h (dimensional)f(system)g(of)g(ordinary)e(di\013eren)n(tial)i(equations)g (called)g(an)g(inertial)523 1511 y(form.)37 b(An)28 b(inertial)f(form)g (can)g(b)r(e)h(regarded)e(as)h(a)g(global)g(\\amplitude")g(equation.) 648 1611 y(This)j(pap)r(er)f(is)h(organized)f(as)g(follo)n(wing.)44 b(In)30 b(sections)g(2,)g(3,)g(w)n(e)g(discuss)g(the)g(prob-)523 1710 y(lem)i(of)g(global)e(existence,)i(uniqueness)g(and)f(regularit)n (y)f(of)i(solutions)f(to)g(the)h(nonlo)r(cal)523 1810 y(equation)23 b(\(1)p Fj(:)p Fs(1\),)h(and)f(consider)f(the)i (dissipativit)n(y)-7 b(,)24 b(i.e.,)g(estimate)f(the)h(size)f(of)g (absorbing)523 1910 y(sets.)49 b(In)33 b(section)e(4,)i(w)n(e)e(pro)n (v)n(e)f(the)j(existence)e(of)h(a)f(global)g(attractor,)h(pro)n(vide)e (upp)r(er)523 2009 y(and)e(lo)n(w)n(er)f(b)r(ound)i(estimates)f(for)g (the)g(dimension)h(of)f(the)h(global)e(attractor,)g(and)h(com-)523 2109 y(pare)i(them)h(with)g(those)f(of)h(the)g(cubic)g(complex)f (Ginzburg-Landau)f(equation.)45 b(In)31 b Fh(x)p Fs(5,)523 2208 y(w)n(e)23 b(sho)n(w)g(the)g(existence)g(of)h(a)f (\014nite-dimensional)g(inertial)g(manifold)g(for)g(this)h(equation,) 523 2308 y(and)31 b(estimate,)h(from)f(ab)r(o)n(v)n(e,)g(its)h (dimension.)47 b(Notice)32 b(that)f(the)h(lo)n(w)n(er)e(b)r(ound)h(for) g(the)523 2408 y(dimension)e(of)g(the)h(global)e(attractor)g(forms)g(a) h(lo)n(w)n(er)f(b)r(ound)h(to)g(the)h(dimension)f(of)g(the)523 2507 y(inertial)j(manifold)g(as)f(w)n(ell.)50 b(W)-7 b(e)32 b(observ)n(e)e(that)j(these)f(b)r(ounds)g(on)f(the)i(dimensions) e(of)523 2607 y(the)f(attractor)e(and)h(of)g(the)h(inertial)f(manifold) g(c)n(hange)g(dramatically)f(as)g(a)h(function)h(of)523 2707 y(the)i(co)r(e\016cien)n(ts)f(of)h(the)g(nonlo)r(cal)e(term.)49 b(In)32 b(section)f(6)77 b(w)n(e)31 b(discuss)g(the)h(concepts)f(of)523 2806 y(determining)i(no)r(des)g(and)h(other)e(degrees)g(of)i(freedom.) 54 b(In)33 b(particular,)h(w)n(e)f(sho)n(w,)h(fol-)523 2906 y(lo)n(wing)d(the)i(w)n(ork)e(of)h(Kuk)-5 b(a)n(vica)30 b([1992],)i(that)g(there)g(are)g(t)n(w)n(o)f(determining)h(no)r(des)g (for)523 3005 y(this)d(nonlo)r(cal)e(equation.)39 b(In)28 b(section)g(7)g(w)n(e)g(consider)f(the)i(sp)r(ecial)f(case)g(of)g(\(1)p Fj(:)p Fs(1\),)g(when)523 3105 y Fj(a)567 3117 y Fr(2)627 3105 y Fs(=)23 b Fj(b)751 3117 y Fr(2)811 3105 y Fs(=)f(0.)35 b(In)21 b(this)g(case)f(w)n(e)h(are)f(able)g(to)h(construct)g (explicitly)g(the)g(Inertial)f(Manifold)523 3205 y(of)28 b(lo)n(w)n(est)e(dimension.)37 b(Finally)27 b(w)n(e)h(end)f(the)h(pap)r (er)g(with)g(a)f(discussion)g(in)h(section)f(8.)523 3479 y Fk(2)135 b(Global)45 b(Existence)h(and)f(Regularit)l(y)i(of)e (Solutions)523 3661 y Fs(W)-7 b(e)20 b(rescale)f(the)h(spatial)f(v)-5 b(ariable)19 b Fj(x)h Fs(to)g(obtain)f(that)i(the)f(p)r(erio)r(dic)f (in)n(terv)-5 b(al)20 b(length)g Fj(l)k Fs(=)f(1.)523 3760 y(In)k(the)g(follo)n(wing,)f(the)i(in)n(tegrals)d(are)h(with)h (resp)r(ect)g(to)g Fj(x)c Fh(2)g Fs([0)p Fj(;)14 b Fs(1])26 b(unless)h(it)g(is)g(sp)r(eci\014ed)523 3860 y(otherwise,)d(and)g Fh(k)12 b(\001)g(k)23 b Fs(denotes)g(the)i Fj(L)1718 3830 y Fr(2)1755 3860 y Fs(-norm.)35 b(F)-7 b(or)23 b(ph)n(ysical)g (reasons)g(w)n(e)h(are)f(in)n(terested)523 3959 y(in)29 b(b)r(ounded)g(solutions,)g(i.e.,)g(solutions)f(in)h Fj(L)1994 3929 y Ff(1)2064 3959 y Fs(.)41 b(Since)29 b(in)g(one)f(spatial)h(dimension)f Fj(L)3317 3929 y Ff(1)523 4059 y Fs(is)37 b(em)n(b)r(edded)h(in)f(the)h(Sob)r(olev)f(space)f Fj(H)1903 4029 y Fr(1)1940 4059 y Fs(,)k(it)e(is)f(su\016cien)n(t,)j (for)d(simplicit)n(y)-7 b(,)40 b(to)d(p)r(ose)523 4159 y(the)e(problem)f(in)h Fj(H)1183 4129 y Fr(1)1255 4159 y Fs(\(see)g(remarks)e(ab)r(out)h(other)g(kinds)h(of)g(regularities)d (later\).)58 b(Let)523 4272 y Fj(A)23 b Fs(=)g Fh(\000)789 4240 y Fi(@)828 4215 y Fe(2)p 771 4254 110 4 v 771 4301 a Fi(@)t(x)848 4284 y Fe(2)910 4272 y Fs(with)d(p)r(erio)r(dic)g(b)r (oundary)f(conditions)h(and)g(let)g Fj(N)9 b Fs(\()p Fj(u)p Fs(\))20 b(denote)g(the)g(remaining)523 4372 y(terms)27 b(of)h(the)g(righ)n(t)f(hand)g(side)h(of)f(\(1)p Fj(:)p Fs(1\).)37 b(Then)28 b(w)n(e)f(can)g(write)h(\(1)p Fj(:)p Fs(1\))f(as)1481 4539 y Fj(du)p 1481 4576 91 4 v 1490 4652 a(dt)1600 4595 y Fs(+)18 b(\()p Fj(a)1759 4607 y Fr(1)1815 4595 y Fs(+)g Fj(ib)1963 4607 y Fr(1)1999 4595 y Fs(\))p Fj(Au)23 b Fs(=)g Fj(N)9 b Fs(\()p Fj(u)p Fs(\))777 b(\(2.1\))523 4799 y(with)28 b(initial)g(data)1671 4898 y Fj(u)p Fs(\()p Fj(x;)14 b Fs(0\))23 b(=)g Fj(u)2068 4910 y Fr(0)2104 4898 y Fs(\()p Fj(x)p Fs(\))p Fj(:)979 b Fs(\(2.2\))1934 5147 y(3)p eop %%Page: 4 4 4 3 bop 523 614 a Fs(Since)27 b(the)g(Sob)r(olev)f(space)g Fj(H)1486 584 y Fr(1)1550 614 y Fs(in)h(one)g(spatial)f(dimension)g(is) h(an)g(algebra,)e(it)i(is)g(easy)e(to)523 714 y(sho)n(w)30 b(that)h Fj(N)37 b Fs(:)29 b Fj(H)1147 684 y Fr(1)1140 735 y Fi(per)1242 714 y Fs(\(0)p Fj(;)14 b Fs(1\))28 b Fh(\000)-14 b(!)29 b Fj(H)1694 684 y Fr(1)1687 735 y Fi(per)1789 714 y Fs(\(0)p Fj(;)14 b Fs(1\))30 b(is)h(a)g(lo)r(cally) f(Lipsc)n(hitz)g(map.)47 b(In)31 b(fact)g(for)523 814 y(an)n(y)c Fj(u;)14 b(v)26 b Fh(2)d Fj(H)985 783 y Fr(1)978 834 y Fi(per)1080 814 y Fs(\(0)p Fj(;)14 b Fs(1\))792 1000 y Fh(k)p Fj(N)9 b Fs(\()p Fj(u)p Fs(\))17 b Fh(\000)h Fj(N)9 b Fs(\()p Fj(v)s Fs(\))p Fh(k)1347 1015 y Fi(H)1405 998 y Fe(1)1526 1000 y Fs(=)82 b Fh(k)p Fj(a)1759 1012 y Fr(0)1796 1000 y Fs(\()p Fj(u)18 b Fh(\000)g Fj(v)s Fs(\))h(+)f(\()p Fj(a)2230 1012 y Fr(2)2286 1000 y Fs(+)g Fj(ib)2434 1012 y Fr(2)2471 1000 y Fs(\)\()p Fh(j)p Fj(u)p Fh(j)2629 966 y Fr(2)2666 1000 y Fj(u)g Fh(\000)g(j)p Fj(v)s Fh(j)2904 966 y Fr(2)2942 1000 y Fj(v)s Fs(\))h(+)1673 1124 y(\()p Fj(a)1749 1136 y Fr(3)1805 1124 y Fs(+)f Fj(ib)1953 1136 y Fr(3)1990 1124 y Fs(\)\()p Fj(u)p Fh(k)p Fj(u)p Fh(k)2234 1090 y Fr(2)2288 1124 y Fh(\000)g Fj(v)s Fh(k)p Fj(v)s Fh(k)2541 1090 y Fr(2)2578 1124 y Fs(\))p Fh(k)2652 1139 y Fi(H)2710 1123 y Fe(1)1526 1249 y Fh(\024)82 b Fj(C)1732 1261 y Fr(1)1770 1249 y Fh(k)p Fj(u)18 b Fh(\000)g Fj(v)s Fh(k)2046 1264 y Fi(H)2104 1247 y Fe(1)2168 1249 y Fj(;)523 1423 y Fs(where)35 b Fj(C)830 1435 y Fr(1)903 1423 y Fs(is)g(a)f(constan)n(t)h(dep)r(ends)h(on)f(\()p Fj(a)1941 1435 y Fr(0)1978 1423 y Fj(;)14 b(a)2059 1435 y Fr(2)2096 1423 y Fj(;)g(b)2169 1435 y Fr(2)2206 1423 y Fj(;)g(a)2287 1435 y Fr(3)2324 1423 y Fj(;)g(b)2397 1435 y Fr(3)2434 1423 y Fj(;)g Fh(k)p Fj(u)p Fh(k)2603 1438 y Fi(H)2661 1421 y Fe(1)2696 1423 y Fj(;)g Fh(k)p Fj(v)s Fh(k)2860 1438 y Fi(H)2918 1421 y Fe(1)2955 1423 y Fs(\).)60 b(Th)n(us)35 b(for)523 1523 y Fj(a)567 1535 y Fr(1)644 1523 y Fj(>)40 b Fs(0)e(the)g(lo)r(cal)f(existence)h(in)g Fj(H)1735 1493 y Fr(1)1810 1523 y Fs(can)f(b)r(e)h(obtained)g(via)f (the)i(semigroup)d(theory)523 1622 y(\(P)n(azy)30 b([1983])f(Theorem)i (6.3.1,)g(or)f(Henry)h([1981])e(Theorem)i(3.3.3\).)46 b(That)31 b(is,)h(if)g Fj(u)3266 1634 y Fr(0)3332 1622 y Fh(2)523 1722 y Fj(H)599 1692 y Fr(1)592 1743 y Fi(per)695 1722 y Fs(\(0)p Fj(;)14 b Fs(1\),)20 b(then)f(there)f(exists)h Fj(T)1588 1692 y Ff(\003)1648 1722 y Fs(=)j Fj(T)1796 1692 y Ff(\003)1834 1722 y Fs(\()p Fh(k)p Fj(u)1956 1734 y Fr(0)1992 1722 y Fh(k)2034 1737 y Fi(H)2092 1720 y Fe(1)2088 1754 y Fd(per)2180 1737 y Fr(\(0)p Fi(;)p Fr(1\))2322 1722 y Fs(\))i Fj(>)e Fs(0)d(suc)n(h)f(that)h(\(2)p Fj(:)p Fs(1\))f(and)h(\(2)p Fj(:)p Fs(2\))523 1838 y(has)27 b(unique)h(lo)r(cal)f(solution)g Fj(u)c Fh(2)g Fj(C)6 b Fs(\([0)p Fj(;)14 b(T)1860 1807 y Ff(\003)1897 1838 y Fs(\);)g Fj(H)2042 1807 y Fr(1)2035 1858 y Fi(per)2138 1838 y Fs(\(0)p Fj(;)g Fs(1\)\))k Fh(\\)h Fj(C)2512 1807 y Fr(1)2550 1838 y Fs(\(\(0)p Fj(;)14 b(T)2754 1807 y Ff(\003)2791 1838 y Fs(\);)g Fj(H)2936 1807 y Fr(1)2929 1858 y Fi(per)3031 1838 y Fs(\(0)p Fj(;)g Fs(1\)\))p Fj(:)648 1948 y Fs(Moreo)n(v)n(er,)31 b(one)h(can)g(also)f(easily)h (sho)n(w)f(that)i Fj(N)40 b Fs(:)31 b Fj(H)2393 1918 y Fi(k)2386 1969 y(per)2488 1948 y Fs(\(0)p Fj(;)14 b Fs(1\))31 b Fh(\000)-14 b(!)31 b Fj(H)2945 1918 y Fi(k)2938 1969 y(per)3040 1948 y Fs(\(0)p Fj(;)14 b Fs(1\))32 b(is)h(a)523 2048 y(lo)r(cally)i(Lipsc)n(hitz)g(map)h(for)f(an)n(y)f(p)r(ositiv)n(e) i(in)n(teger)e Fj(k)s Fs(.)61 b(Therefore,)36 b(within)g(the)g(short) 523 2147 y(time)i(of)f(existence)g(0)i Fj(<)g(t)g(<)f(T)1607 2117 y Ff(\003)1645 2147 y Fs(,)i(w)n(e)c(can)h(also)g(sho)n(w)f(that)h (the)h(solution)f(is)g(indeed)523 2247 y(b)r(elongs)26 b(to)g Fj(H)996 2217 y Fi(k)989 2268 y(per)1092 2247 y Fs(\(0)p Fj(;)14 b Fs(1\))26 b(for)f(an)n(y)h(p)r(ositiv)n(e)g(in)n (terger)f Fj(k)k Fs(\(see)e(Henry)f([1981],)f(page)g(73\).)36 b(In)523 2347 y(particular,)e(for)f(an)n(y)h(0)e Fj(<)h(t)h(<)f(T)1629 2317 y Ff(\003)1700 2347 y Fs(the)h(solution)g(b)r(elongs)f(to)g Fj(C)2646 2317 y Ff(1)2717 2347 y Fs(\(0)p Fj(;)14 b Fs(1\).)55 b(F)-7 b(or)33 b(further)523 2446 y(discussion)c(concerning) f(the)i(regularit)n(y)e(of)h(the)h(solution)f(see)g(the)h(Remark)f(at)g (the)h(end)523 2546 y(of)e(this)f(section.)648 2646 y(First)32 b(observ)n(e)f(that)i(when)g Fj(a)1607 2658 y Fr(2)1666 2646 y Fs(+)22 b Fj(a)1797 2658 y Fr(3)1865 2646 y Fj(>)32 b Fs(0)g(there)g(are)g(solutions)g(whic)n(h)h(blo)n(w)f(up)h(in)523 2745 y(\014nite)38 b(time.)68 b(Indeed,)40 b(let)e(us)f(seek)g(a)g (solution)h(to)f(the)h(equation)f(\(1.1\))g(of)h(the)g(form)523 2845 y Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))24 b(=)e Fj(U)9 b Fs(\()p Fj(t)p Fs(\).)37 b(Hence,)28 b Fj(U)9 b Fs(\()p Fj(t)p Fs(\))28 b(satis\014es)1244 2998 y(_)1220 3019 y Fj(U)k Fs(=)23 b Fj(a)1441 3031 y Fr(0)1478 3019 y Fj(U)k Fs(+)18 b([\()p Fj(a)1744 3031 y Fr(2)1800 3019 y Fs(+)g Fj(a)1927 3031 y Fr(3)1964 3019 y Fs(\))h(+)f Fj(i)p Fs(\()p Fj(b)2195 3031 y Fr(2)2250 3019 y Fs(+)g Fj(b)2369 3031 y Fr(3)2406 3019 y Fs(\)])p Fh(j)p Fj(U)9 b Fh(j)2573 2985 y Fr(2)2610 3019 y Fj(U:)523 3193 y Fs(Let)28 b(us)f(m)n(ultiply)h(the)g(ab)r(o)n(v)n(e)f(equation)g(b)n(y) 1965 3172 y(\026)1943 3193 y Fj(A)h Fs(and)f(tak)n(e)g(the)h(real)f (part)g(to)g(get)1342 3350 y(1)p 1342 3387 42 4 v 1342 3463 a(2)1418 3350 y Fj(d)p 1403 3387 74 4 v 1403 3463 a(dt)1486 3406 y Fh(j)p Fj(U)9 b Fh(j)1598 3371 y Fr(2)1659 3406 y Fs(=)22 b Fj(a)1790 3418 y Fr(0)1827 3406 y Fh(j)p Fj(U)9 b Fh(j)1939 3371 y Fr(2)1995 3406 y Fs(+)18 b(\()p Fj(a)2154 3418 y Fr(2)2210 3406 y Fs(+)g Fj(a)2337 3418 y Fr(3)2374 3406 y Fs(\))p Fh(j)p Fj(U)9 b Fh(j)2518 3371 y Fr(4)2555 3406 y Fj(:)639 b Fs(\(2.3\))523 3603 y(It)37 b(is)f(clear)f(that)h(certain)g(solutions)f(to)h(this)h (equation)e(will)i(blo)n(w)e(up)i(in)f(\014nite)h(time,)523 3703 y(pro)n(vided)i(that)i Fj(a)1114 3715 y Fr(2)1178 3703 y Fs(+)27 b Fj(a)1314 3715 y Fr(3)1396 3703 y Fj(>)44 b Fs(0.)75 b(Therefore)39 b(a)i(necessary)d(condition)j(for)f(the)h (global)523 3802 y(existence)35 b(is)g Fj(a)1019 3814 y Fr(2)1079 3802 y Fs(+)23 b Fj(a)1211 3814 y Fr(3)1284 3802 y Fh(\024)35 b Fs(0.)59 b(Using)35 b(argumen)n(t)f(similar)g(to)h (the)h(ones)e(b)r(elo)n(w)h(one)g(can)523 3902 y(sho)n(w)h(the)h (global)e(existence)h(of)h(solution)f(for)g(the)h(case)f Fj(a)2449 3914 y Fr(2)2510 3902 y Fs(+)24 b Fj(a)2643 3914 y Fr(3)2718 3902 y Fs(=)38 b(0.)63 b(Ho)n(w)n(ev)n(er,)37 b(in)523 4002 y(this)26 b(case)f(one)g(can)g(see)g(from)g(\(2.3\))h (that)g(all)f(spatially)g(homogeneous)f(solutions)g(except)523 4101 y(zero)h(will)i(gro)n(w)e(exp)r(onen)n(tially)h(for)g Fj(a)1734 4113 y Fr(0)1794 4101 y Fj(>)d Fs(0)j(and)g(hence)h(the)g (system)f(is)h(not)f(dissipativ)n(e.)523 4201 y(Therefore)k(w)n(e)g (will)i(concen)n(trate)d(on)i(the)g(case)g(for)f Fj(a)2256 4213 y Fr(2)2314 4201 y Fs(+)20 b Fj(a)2443 4213 y Fr(3)2509 4201 y Fj(<)28 b Fs(0)j(from)g(no)n(w)f(on)h(unless)523 4301 y(otherwise)22 b(sp)r(eci\014ed.)36 b(By)23 b(assuming)f(further)h (that)h Fj(a)2231 4313 y Fr(2)2291 4301 y Fj(<)f Fs(0,)h(w)n(e)e(are)h (going)f(to)h(sho)n(w)f(that)523 4400 y(the)27 b(suprem)n(um)g(of)g (the)g Fj(H)1381 4370 y Fr(1)1418 4400 y Fs(-norm)f(of)h(the)g (solution)g(remains)f(b)r(ounded)h(uniformly)g(o)n(v)n(er)523 4500 y(an)n(y)j(in)n(terv)-5 b(al)30 b(of)h(time)g(for)g(whic)n(h)f (the)i(solution)e(exists.)46 b(F)-7 b(rom)30 b(this)i(w)n(e)e(infer)h (that)g(the)523 4599 y(solution)g(exists)h(for)f(all)h Fj(t)e(>)g Fs(0.)50 b(Supp)r(ose)32 b(that)g(the)g(solution)f(exists)h (on)g(the)g(maximal)523 4699 y(in)n(terv)-5 b(al)31 b(of)h(existence)g ([0)p Fj(;)14 b(T)1435 4711 y Fr(max)1560 4699 y Fs(\).)50 b(W)-7 b(e)32 b(need)g(to)g(sho)n(w)f(that)h Fj(T)2559 4711 y Fr(max)2715 4699 y Fs(=)e Fh(1)p Fs(.)50 b(T)-7 b(o)31 b(this)h(end)523 4799 y(it)27 b(is)f(enough)g(to)g(sho)n(w)g (that)h(the)f Fj(H)1673 4769 y Fr(1)1710 4799 y Fs(-norm)g(of)g(the)h (solution)f(is)h(b)r(ounded)f(uniformly)h(on)523 4898 y(the)k(in)n(terv)-5 b(al)30 b([0)p Fj(;)14 b(T)1123 4910 y Fr(max)1249 4898 y Fs(\).)47 b(This)30 b(will)h(b)r(e)h(done)e (b)n(y)g(straigh)n(t)g(forw)n(ard)f(energy)h(estimates.)1934 5147 y(4)p eop %%Page: 5 5 5 4 bop 523 614 a Fs(Let)34 b(us)f(\014rst)h(establish)f(the)h(uniform) g(b)r(ound)g(for)f(the)h Fj(L)2384 584 y Fr(2)2421 614 y Fs(-norm)e(of)i Fj(u)p Fs(.)55 b(T)-7 b(ak)n(e)32 b(the)i(real)523 714 y(part)27 b(of)h(the)g Fj(L)998 684 y Fr(2)1062 714 y Fs(inner)g(pro)r(duct)f(of)h(the)g(equation)f(\(1)p Fj(:)p Fs(1\))g(with)33 b(\026)-47 b Fj(u)27 b Fs(to)h(obtain)876 900 y(1)p 876 937 42 4 v 876 1013 a(2)952 900 y Fj(d)p 937 937 74 4 v 937 1013 a(dt)1020 956 y Fh(k)p Fj(u)p Fh(k)1152 922 y Fr(2)1211 956 y Fs(=)23 b Fj(a)1343 968 y Fr(0)1380 956 y Fh(k)p Fj(u)p Fh(k)1512 922 y Fr(2)1566 956 y Fh(\000)18 b Fj(a)1693 968 y Fr(1)1730 956 y Fh(k)p Fj(u)1820 968 y Fi(x)1861 956 y Fh(k)1903 922 y Fr(2)1958 956 y Fs(+)g Fj(a)2085 968 y Fr(2)2136 843 y Fg(Z)2219 864 y Fr(1)2183 1032 y(0)2271 956 y Fh(j)p Fj(u)p Fs(\()p Fj(x;)c(t)p Fs(\))p Fh(j)2543 922 y Fr(4)2580 956 y Fj(dx)19 b Fs(+)f Fj(a)2816 968 y Fr(3)2854 956 y Fh(k)p Fj(u)p Fh(k)2986 922 y Fr(4)3021 956 y Fj(:)173 b Fs(\(2.4\))523 1191 y(Since)28 b(w)n(e)f(assumed)g Fj(a)1239 1203 y Fr(2)1299 1191 y Fj(<)c Fs(0)k(then)h Fj(a)1689 1203 y Fr(2)1726 1191 y Fh(k)p Fj(u)p Fh(k)1858 1160 y Fr(4)1858 1217 y Fi(L)1904 1200 y Fe(4)1962 1191 y Fh(\024)23 b Fj(a)2094 1203 y Fr(2)2131 1191 y Fh(k)p Fj(u)p Fh(k)2263 1160 y Fr(4)2263 1217 y Fi(L)2309 1200 y Fe(2)2371 1191 y Fs(and)28 b(\(2)p Fj(:)p Fs(4\))f(b)r(ecomes)1118 1366 y(1)p 1118 1403 42 4 v 1118 1479 a(2)1194 1366 y Fj(d)p 1179 1403 74 4 v 1179 1479 a(dt)1262 1422 y Fh(k)p Fj(u)p Fh(k)1394 1388 y Fr(2)1453 1422 y Fh(\024)c Fj(a)1585 1434 y Fr(0)1622 1422 y Fh(k)p Fj(u)p Fh(k)1754 1388 y Fr(2)1808 1422 y Fh(\000)18 b Fj(a)1935 1434 y Fr(1)1972 1422 y Fh(k)p Fj(u)2062 1434 y Fi(x)2103 1422 y Fh(k)2145 1388 y Fr(2)2200 1422 y Fs(+)g(\()p Fj(a)2359 1434 y Fr(2)2415 1422 y Fs(+)g Fj(a)2542 1434 y Fr(3)2579 1422 y Fs(\))p Fh(k)p Fj(u)p Fh(k)2743 1388 y Fr(4)2779 1422 y Fj(:)415 b Fs(\(2.5\))523 1628 y(Note)29 b(that)g(since)g(w)n(e)g (assumed)f Fj(a)1613 1640 y Fr(1)1675 1628 y Fj(>)d Fs(0)k(and)g Fj(a)2043 1640 y Fr(2)2099 1628 y Fs(+)19 b Fj(a)2227 1640 y Fr(3)2289 1628 y Fj(<)25 b Fs(0,)k(hereafter)f(w)n(e)h(assume)f (that)523 1728 y Fj(a)567 1740 y Fr(0)627 1728 y Fj(>)23 b Fs(0)k(otherwise)g(the)h(dynamics)f(is)g(trivial.)37 b(Using)27 b(Y)-7 b(oung's)27 b(inequalit)n(y)1275 1959 y Fj(a)1319 1971 y Fr(0)1356 1959 y Fh(k)p Fj(u)p Fh(k)1488 1925 y Fr(2)1547 1959 y Fh(\024)22 b(j)p Fj(a)1701 1971 y Fr(2)1757 1959 y Fs(+)c Fj(a)1884 1971 y Fr(3)1921 1959 y Fh(j)28 b(k)p Fj(u)p Fh(k)2104 1925 y Fr(4)2158 1959 y Fs(+)2386 1903 y Fj(a)2430 1873 y Fr(2)2430 1923 y(0)p 2251 1940 352 4 v 2251 2016 a Fs(4)p Fh(j)p Fj(a)2360 2028 y Fr(2)2415 2016 y Fs(+)18 b Fj(a)2542 2028 y Fr(3)2579 2016 y Fh(j)2612 1959 y Fj(;)523 2186 y Fs(and)27 b(so)g(w)n(e)h(ha)n (v)n(e)1355 2262 y(1)p 1355 2299 42 4 v 1355 2375 a(2)1432 2262 y Fj(d)p 1417 2299 74 4 v 1417 2375 a(dt)1500 2318 y Fh(k)p Fj(u)p Fh(k)1632 2283 y Fr(2)1686 2318 y Fs(+)18 b Fj(a)1813 2330 y Fr(1)1850 2318 y Fh(k)p Fj(u)1940 2330 y Fi(x)1981 2318 y Fh(k)2023 2283 y Fr(2)2083 2318 y Fh(\024)2316 2262 y Fj(a)2360 2231 y Fr(2)2360 2282 y(0)p 2181 2299 352 4 v 2181 2375 a Fs(4)p Fh(j)p Fj(a)2290 2387 y Fr(2)2345 2375 y Fs(+)g Fj(a)2472 2387 y Fr(3)2509 2375 y Fh(j)2542 2318 y Fj(:)652 b Fs(\(2.6\))523 2548 y(Let)24 b Fj(K)739 2560 y Fr(0)798 2548 y Fs(=)992 2505 y Fi(a)1028 2480 y Fe(2)1028 2522 y(0)p 896 2529 261 4 v 896 2576 a Fr(4)p Ff(j)p Fi(a)985 2584 y Fe(2)1017 2576 y Fr(+)p Fi(a)1104 2584 y Fe(3)1137 2576 y Ff(j)1166 2548 y Fs(,)h(if)e(w)n(e)g(ignore)f(the)i(term)g Fj(a)2026 2560 y Fr(1)2063 2548 y Fh(k)p Fj(u)2153 2560 y Fi(x)2193 2548 y Fh(k)2235 2518 y Fr(2)2296 2548 y Fs(on)f(the)g(left)h(hand)g (side,)g(equation)523 2661 y(\(2)p Fj(:)p Fs(6\))j(giv)n(es)g(us)1421 2761 y Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)1647 2727 y Fr(2)1706 2761 y Fh(\024)c(k)p Fj(u)1884 2773 y Fr(0)1920 2761 y Fh(k)1962 2727 y Fr(2)2017 2761 y Fs(+)18 b(2)p Fj(K)2213 2773 y Fr(0)2250 2761 y Fj(T)2299 2773 y Fr(max)2453 2761 y Fj(;)741 b Fs(\(2.7\))523 2910 y(for)29 b(ev)n(ery)f Fj(t)f Fh(2)f Fs([0)p Fj(;)14 b(T)1162 2922 y Fr(max)1288 2910 y Fs(\).)43 b(Moreo)n(v)n(er)27 b(b)n(y)i(in)n(tegrating)g(equation)g(\(2)p Fj(:)p Fs(6\))g(with)h (resp)r(ect)f(to)523 3010 y Fj(\034)37 b Fs(o)n(v)n(er)26 b(\()p Fj(s;)14 b(t)p Fs(\))24 b Fh(\032)e Fs([0)p Fj(;)14 b(T)1206 3022 y Fr(max)1332 3010 y Fs(\),)28 b(w)n(e)f(obtain)982 3194 y(1)p 982 3231 42 4 v 982 3307 a(2)1034 3250 y Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)1260 3216 y Fr(2)1315 3250 y Fh(\000)1408 3194 y Fs(1)p 1408 3231 V 1408 3307 a(2)1459 3250 y Fh(k)p Fj(u)p Fs(\()p Fj(s)p Fs(\))p Fh(k)1694 3216 y Fr(2)1749 3250 y Fs(+)18 b Fj(a)1876 3262 y Fr(1)1927 3137 y Fg(Z)2010 3158 y Fi(t)1973 3326 y(s)2053 3250 y Fh(k)p Fj(u)2143 3262 y Fi(x)2184 3250 y Fs(\()p Fj(\034)9 b Fs(\))p Fh(k)2335 3216 y Fr(2)2373 3250 y Fj(d\034)33 b Fh(\024)22 b Fj(K)2643 3262 y Fr(0)2680 3250 y Fs(\()p Fj(t)d Fh(\000)f Fj(s)p Fs(\))p Fj(:)523 3480 y Fs(Consequen)n(tly)27 b(b)n(y)g(using)g(\(2)p Fj(:)p Fs(7\))h(w)n(e)f(ha)n(v)n(e,)1025 3724 y Fj(a)1069 3736 y Fr(1)1120 3611 y Fg(Z)1203 3631 y Fi(T)1242 3639 y Fe(max)1166 3800 y Fr(0)1368 3724 y Fh(k)p Fj(u)1458 3736 y Fi(x)1498 3724 y Fs(\()p Fj(\034)9 b Fs(\))p Fh(k)1649 3690 y Fr(2)1687 3724 y Fj(d\034)33 b Fh(\024)1897 3668 y Fs(1)p 1897 3705 V 1897 3781 a(2)1949 3724 y Fh(k)p Fj(u)2039 3736 y Fr(0)2075 3724 y Fh(k)2117 3690 y Fr(2)2172 3724 y Fs(+)18 b(2)p Fj(K)2368 3736 y Fr(0)2404 3724 y Fj(T)2453 3736 y Fr(max)2603 3724 y Fs(:=)k Fj(K)2784 3736 y Fr(1)2849 3724 y Fj(;)345 b Fs(\(2.8\))523 3953 y(where)26 b Fj(K)833 3965 y Fr(1)897 3953 y Fs(dep)r(ends)i(on)e Fh(k)p Fj(u)1422 3965 y Fr(0)1459 3953 y Fh(k)g Fs(and)h Fj(T)1737 3965 y Fr(max)1863 3953 y Fs(.)37 b(W)-7 b(e)27 b(no)n(w)g(establish)f(the)i(uniform)f(b)r(ound)g(for)523 4053 y Fh(jj)p Fj(u)617 4065 y Fi(x)658 4053 y Fh(jj)p Fs(.)51 b(Based)31 b(on)h(our)g(previous)e(remarks)h(ab)r(out)h(the)g (regularit)n(y)f(of)h(the)g(solution)g(w)n(e)523 4152 y(kno)n(w)23 b(that)g(the)h(solution)f(is)g(at)g(least)g(in)g Fj(H)1894 4122 y Fr(2)1931 4152 y Fs(,)i(for)d Fj(t)h Fh(2)h Fs(\(0)p Fj(;)14 b(T)2393 4164 y Fr(max)2519 4152 y Fs(\).)36 b(W)-7 b(e)23 b(m)n(ultiply)h(equation)523 4252 y(\(1)p Fj(:)p Fs(1\))i(with)32 b(\026)-47 b Fj(u)956 4264 y Fi(xx)1035 4252 y Fs(,)27 b(in)n(tegrate)e(with)i(resp)r(ect)f (to)g Fj(x)h Fs(o)n(v)n(er)e(the)i(in)n(terv)-5 b(al)25 b([0)p Fj(;)14 b Fs(1])26 b(and)g(then)h(tak)n(e)523 4352 y(the)h(real)f(part)g(to)g(get)889 4533 y(1)p 889 4570 V 889 4646 a(2)965 4533 y Fj(d)p 950 4570 74 4 v 950 4646 a(dt)1034 4589 y Fh(k)p Fj(u)1124 4601 y Fi(x)1164 4589 y Fh(k)1206 4555 y Fr(2)1326 4589 y Fs(=)83 b Fj(a)1518 4601 y Fr(0)1555 4589 y Fh(k)p Fj(u)1645 4601 y Fi(x)1686 4589 y Fh(k)1728 4555 y Fr(2)1783 4589 y Fh(\000)18 b Fj(a)1910 4601 y Fr(1)1947 4589 y Fh(k)p Fj(u)2037 4601 y Fi(xx)2116 4589 y Fh(k)2158 4555 y Fr(2)2213 4589 y Fs(+)g(2)p Fj(a)2382 4601 y Fr(2)2432 4476 y Fg(Z)2515 4497 y Fr(1)2478 4665 y(0)2566 4589 y Fh(j)p Fj(u)p Fh(j)2660 4555 y Fr(2)2697 4589 y Fh(j)p Fj(u)2768 4601 y Fi(x)2810 4589 y Fh(j)2833 4555 y Fr(2)2870 4589 y Fj(dx)1474 4828 y Fs(+)p Fj(R)q(e)p Fs([\()p Fj(a)1741 4840 y Fr(2)1796 4828 y Fs(+)g Fj(ib)1944 4840 y Fr(2)1980 4828 y Fs(\))2026 4715 y Fg(Z)2109 4735 y Fr(1)2072 4903 y(0)2160 4828 y Fj(u)2208 4794 y Fr(2)2250 4828 y Fs(\026)-47 b Fj(u)2293 4794 y Fr(2)2293 4848 y Fi(x)2334 4828 y Fj(dx)p Fs(])20 b(+)e Fj(a)2594 4840 y Fr(3)2631 4828 y Fh(k)p Fj(u)p Fh(k)2763 4794 y Fr(2)2799 4828 y Fh(k)p Fj(u)2889 4840 y Fi(x)2929 4828 y Fh(k)2971 4794 y Fr(2)3008 4828 y Fj(:)186 b Fs(\(2.9\))1934 5147 y(5)p eop %%Page: 6 6 6 5 bop 523 614 a Fs(By)27 b(using)h(a)f(series)f(of)i(Agmon's)f(and)g (Y)-7 b(oung's)27 b(inequalities,)983 717 y Fg(Z)1066 738 y Fr(1)1029 906 y(0)1117 830 y Fh(j)p Fj(u)p Fh(j)1211 796 y Fr(2)1248 830 y Fh(j)p Fj(u)1319 842 y Fi(x)1361 830 y Fh(j)1384 796 y Fr(2)1444 830 y Fh(\024)c(k)p Fj(u)p Fh(k)1664 796 y Fr(2)1664 851 y Fi(L)1710 834 y Fc(1)1772 830 y Fh(k)p Fj(u)1862 842 y Fi(x)1903 830 y Fh(k)1945 796 y Fr(2)2005 830 y Fh(\024)2103 774 y Fs(3)p 2103 811 42 4 v 2103 887 a(2)2154 830 y Fh(k)p Fj(u)p Fh(k)2286 796 y Fr(2)2322 830 y Fh(k)p Fj(u)2412 842 y Fi(x)2453 830 y Fh(k)2495 796 y Fr(2)2550 830 y Fs(+)2643 774 y(1)p 2643 811 V 2643 887 a(2)2694 830 y Fh(k)p Fj(u)2784 842 y Fi(x)2825 830 y Fh(k)2867 796 y Fr(4)2904 830 y Fj(:)248 b Fs(\(2.10\))523 1037 y(Because)26 b(of)h(the)g(uniform)g(b)r(ound)g (\(2)p Fj(:)p Fs(7\),)g(equation)g(\(2)p Fj(:)p Fs(9\))f(can)h(b)r(e)g (estimated)g(as)f(follo)n(ws)1262 1185 y Fj(d)p 1247 1222 74 4 v 1247 1299 a(dt)1330 1242 y Fh(k)p Fj(u)1420 1254 y Fi(x)1461 1242 y Fh(k)1503 1207 y Fr(2)1558 1242 y Fs(+)18 b(2)p Fj(a)1727 1254 y Fr(1)1764 1242 y Fh(k)p Fj(u)1854 1254 y Fi(xx)1933 1242 y Fh(k)1975 1207 y Fr(2)2034 1242 y Fh(\024)23 b Fj(C)2181 1254 y Fr(2)2219 1242 y Fs(\(1)18 b(+)g Fh(k)p Fj(u)2484 1254 y Fi(x)2525 1242 y Fh(k)2567 1207 y Fr(2)2604 1242 y Fs(\))2636 1207 y Fr(2)3175 1242 y Fs(\(2.11\))523 1436 y(where)26 b Fj(C)821 1448 y Fr(2)882 1436 y Fs(=)d Fj(C)1029 1448 y Fr(2)1066 1436 y Fs(\()p Fj(a)1142 1448 y Fr(0)1180 1436 y Fj(;)14 b(a)1261 1448 y Fr(2)1298 1436 y Fj(;)g(b)1371 1448 y Fr(2)1407 1436 y Fj(;)g(a)1488 1448 y Fr(3)1525 1436 y Fj(;)g Fh(k)p Fj(u)1652 1448 y Fr(0)1689 1436 y Fh(k)p Fj(;)g(T)1817 1448 y Fr(max)1942 1436 y Fs(\).)37 b(Let)27 b Fj(y)f Fs(:=)d(1)16 b(+)g Fh(k)p Fj(u)2589 1448 y Fi(x)2630 1436 y Fh(k)2672 1406 y Fr(2)2736 1436 y Fs(so)26 b(equation)g(\(2)p Fj(:)p Fs(11\))523 1535 y(b)r(ecomes)1756 1601 y Fj(dy)p 1756 1638 87 4 v 1763 1714 a(dt)1876 1657 y Fh(\024)d Fj(C)2023 1669 y Fr(2)2060 1657 y Fj(y)2104 1623 y Fr(2)2141 1657 y Fj(:)1011 b Fs(\(2.12\))523 1822 y(In)28 b(addition)f(based)g (on)h(\(2)p Fj(:)p Fs(8\))f(w)n(e)g(ha)n(v)n(e)1330 1931 y Fg(Z)1413 1952 y Fi(T)1452 1960 y Fe(max)1376 2120 y Fr(0)1578 2044 y Fj(y)s Fs(\()p Fj(\034)9 b Fs(\))p Fj(d\034)34 b Fh(\024)22 b Fj(T)1980 2056 y Fr(max)2125 2044 y Fs(+)c Fj(K)2279 2056 y Fr(1)2339 2044 y Fs(:=)23 b Fj(K)2521 2056 y Fr(2)2557 2044 y Fj(:)595 b Fs(\(2.13\))523 2252 y(By)31 b(a)g(generalized)f(v)n(ersion)f(of)j(the)f(Gron)n(w)n (all's)f(inequalit)n(y)h(\(see,)h(for)f(instance,)h(F)-7 b(oias-)523 2351 y(Sell-T)g(emam)27 b([1988])f(and)i(T)-7 b(emam)27 b([1988]\))f(equations)h(\(2)p Fj(:)p Fs(12\))g(and)g(\(2)p Fj(:)p Fs(13\))g(imply)h(that)1745 2512 y Fj(y)s Fs(\()p Fj(t)p Fs(\))23 b Fh(\024)f Fj(K)2064 2524 y Fr(3)2129 2512 y Fj(;)1023 b Fs(\(2.14\))523 2673 y(for)32 b(some)g(p)r(ositiv)n (e)g(constan)n(t)g Fj(K)1591 2685 y Fr(3)1660 2673 y Fs(whic)n(h)g(dep)r(ends)h(on)g Fh(k)p Fj(u)2439 2685 y Fr(0)2475 2673 y Fh(k)f Fs(and)g Fj(T)2764 2685 y Fr(max)2890 2673 y Fs(.)52 b(Hence)33 b(\(2)p Fj(:)p Fs(7\))523 2773 y(and)k(\(2)p Fj(:)p Fs(14\))f(imply)i(that)f Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)1601 2788 y Fi(H)1659 2771 y Fe(1)1655 2804 y Fd(per)1747 2788 y Fr(\(0)p Fi(;)p Fr(1\))1926 2773 y Fs(is)g(uniformly)g(b)r(ounded)h(for)e Fj(t)j Fh(2)g Fs([0)p Fj(;)14 b(T)3206 2785 y Fr(max)3332 2773 y Fs(\).)523 2883 y(Therefore)24 b(w)n(e)h(can)g(extend)h(the)g (solution)f(to)g(an)n(y)g Fj(t)e(>)g Fs(0)i(\(P)n(azy)f([1983]\),)g (and)i(w)n(e)f(obtain)523 2983 y(the)g(global)f(existence)g(of)g(mild)i (solutions)d(in)i Fj(H)2046 2952 y Fr(1)2083 2983 y Fs(.)36 b(Th)n(us)25 b(w)n(e)f(ha)n(v)n(e)f(pro)n(v)n(ed)g(the)i(follo)n(wing) 523 3082 y(theorem:)523 3243 y Fo(Theorem)30 b(1)42 b Fn(Assume)23 b(that)h Fj(a)1530 3255 y Fr(1)1590 3243 y Fj(>)f Fs(0)p Fj(;)14 b(a)1801 3255 y Fr(2)1860 3243 y Fh(\024)23 b Fs(0)g Fn(and)i Fj(a)2213 3255 y Fr(2)2256 3243 y Fs(+)6 b Fj(a)2371 3255 y Fr(3)2431 3243 y Fh(\024)22 b Fs(0)p Fn(.)37 b(Then)24 b(for)h(every)g(initial)523 3343 y(data)k Fj(u)754 3355 y Fr(0)814 3343 y Fh(2)23 b Fj(H)968 3313 y Fr(1)961 3363 y Fi(per)1063 3343 y Fs(\(0)p Fj(;)14 b Fs(1\))28 b Fn(ther)l(e)g(exists)f(a)h(unique)g (solution)g Fj(u)f Fn(to)h(the)g(pr)l(oblem)g(\(1.1\)-\(1.2\)-)523 3442 y(\(1.3\),)j(which)h(is)e(de\014ne)l(d)g(glob)l(al)t(ly)h(for)g (al)t(l)g Fj(t)23 b(>)f Fs(0)p Fn(.)39 b(Mor)l(e)l(over,)31 b(this)f(solution)g(is)g(analytic)523 3542 y(in)g(sp)l(ac)l(e)h(and)f (time)h(for)f(al)t(l)i Fj(t)23 b(>)g Fs(0)p Fn(.)39 b(If)31 b(in)f(addition)i(we)e(assume)g Fj(a)2665 3554 y Fr(2)2721 3542 y Fs(+)18 b Fj(a)2848 3554 y Fr(3)2909 3542 y Fj(<)23 b Fs(0)p Fn(,)30 b(then)g(the)523 3642 y(solution)g(is)g(b)l(ounde)l(d) g(uniformly)h(in)f(time.)648 3802 y Fo(Remark.)40 b Fs(Alternativ)n (ely)-7 b(,)29 b(one)g(can)g(use)g(the)h(Galerkin)e(pro)r(cedure)g(to)i (pro)n(v)n(e)d(exis-)523 3902 y(tence)32 b(and)g(regularit)n(y)e(of)i (solutions.)50 b(Actually)-7 b(,)33 b(under)f(the)h(same)e(condition,)i (w)n(e)f(can)523 4002 y(sho)n(w)k(that)i(w)n(eak)e(solutions)g(exist)h (for)g Fj(u)1892 4014 y Fr(0)1967 4002 y Fh(2)j Fj(L)2119 3972 y Fr(2)2119 4022 y Fi(per)2221 4002 y Fs(\(0)p Fj(;)14 b Fs(1\))36 b(via)h(the)g(Galerkin)g(metho)r(d.)523 4101 y(In)32 b(particular,)g(follo)n(wing)f(the)h(w)n(ork)f(of)h(F)-7 b(errari)30 b(and)i(Titi)g([1994],)f(whic)n(h)h(is)g(based)g(on)523 4201 y(F)-7 b(oias)31 b(and)g(T)-7 b(emam)32 b([1989],)f(one)g(can)g (sho)n(w)g(that)h(the)g(solution)f(b)r(elongs)g(to)h(a)f(Gevrey)523 4301 y(class)e(of)g(regularit)n(y)f(and)h(as)g(a)h(result)f(it)h(is)f (real)g(analytic)g(with)h(resp)r(ect)g(to)f(the)h(spatial)523 4400 y(v)-5 b(ariable)32 b Fj(x)h Fs(\(see)g(also)f(Do)r(elman)h(and)g (Titi)g([1993],)g(Duan,)h(Holmes)f(and)f(Titi)i([1993]\).)523 4500 y(Moreo)n(v)n(er,)22 b(one)h(can)g(also)f(implemen)n(t)i(the)g (approac)n(h)d(of)j(F)-7 b(oias)22 b(and)h(T)-7 b(emam)24 b([1979])d(\(see)523 4599 y(also)h(Constan)n(tin)h(and)h(F)-7 b(oias)23 b([1988])e(and)j(F)-7 b(oias)22 b(and)i(T)-7 b(emam)23 b([1989]\))f(to)i(sho)n(w)e(that)i(the)523 4699 y(solution)k(is)g(analytic)f(with)i(resp)r(ect)e(to)h(the)h(temp)r (oral)e(v)-5 b(ariable)27 b Fj(t)i Fs(as)e(w)n(ell.)38 b(F)-7 b(or)27 b(further)523 4799 y(results)i(concerning)g(the)h (existence)g(uniqueness)g(and)g(regularit)n(y)e(of)i(a)f(class)g(of)h (\\lo)r(cal")523 4898 y(complex)d(Ginzburg-Landau)f(equations)h(see)g (Lev)n(ermore)f(and)h(Oliv)n(er)g([1995].)1934 5147 y(6)p eop %%Page: 7 7 7 6 bop 523 614 a Fk(3)135 b(Radii)46 b(of)f(the)g(absorbing)g(balls) 523 796 y Fs(>F)-7 b(rom)36 b(no)n(w)g(on)g(w)n(e)h(will)f(assume)g (that)h Fj(a)1921 808 y Fr(2)1983 796 y Fs(+)23 b Fj(a)2115 808 y Fr(3)2191 796 y Fj(<)37 b Fs(0)f(and)h Fj(a)2586 808 y Fr(2)2647 796 y Fs(+)24 b Fj(ib)2801 808 y Fr(2)2875 796 y Fh(6)p Fs(=)38 b(0.)63 b(W)-7 b(e)37 b(will)523 896 y(discuss)26 b(in)h(full)h(details)e(the)h(case)f(when)h Fj(a)1880 908 y Fr(2)1940 896 y Fs(=)22 b Fj(b)2063 908 y Fr(2)2123 896 y Fs(=)h(0)j(in)h(section)g(7.)36 b(In)27 b(this)f(section)h(w)n(e)523 995 y(will)h(follo)n(w)f(the)h(w)n(ork)f (of)h(Do)r(ering)f(et)h(al)f([1988])g(to)g(get)h(estimates)f(for)h(the) g(radii)f(of)h(the)523 1095 y(absorbing)19 b(balls.)35 b(Moreo)n(v)n(er)18 b(w)n(e)j(will)g(sho)n(w)g(that)g(our)f(estimates)h (agree)f(with)h(that)h(of)f(the)523 1195 y(cubic)32 b(Ginzburg-Landau)e (equation)h(in)h(Do)r(ering)g(et)g(al.)49 b([1988])29 b(when)j(the)h(co)r(e\016cien)n(t)523 1294 y(of)g(the)h(nonlo)r(cal)e (term)i(is)f(zero.)53 b(In)33 b(the)h(previous)e(section)h Fh(x)p Fs(2,)h(when)g(w)n(e)f(established)523 1394 y(global)c (existence,)i(our)f(estimate)g(for)g(the)h Fj(L)1969 1364 y Fr(2)2005 1394 y Fs(-norm)f(w)n(as)f(rough.)44 b(W)-7 b(e)31 b(will)g(no)n(w)e(giv)n(e)h(a)523 1494 y(sharp)i(estimate)h(for)f(the)h(size)g(of)f(the)h(absorbing)e(ball)i (in)g Fj(L)2493 1463 y Fr(2)2563 1494 y Fs(\(see)f(the)i(expression)d (for)523 1593 y(the)26 b(Stok)n(es)e(solution)g(b)r(elo)n(w)h (\(4.26\)\).)36 b(By)25 b(dropping)f(the)h(term)g Fh(\000)p Fj(a)2689 1605 y Fr(1)2726 1593 y Fh(k)p Fj(u)2816 1605 y Fi(x)2857 1593 y Fh(k)2899 1563 y Fr(2)2961 1593 y Fs(on)g(the)g(righ)n(t)523 1693 y(hand)j(side)f(of)h(equation)f(\(2)p Fj(:)p Fs(5\),)g(w)n(e)g(ha)n(v)n(e)1128 1861 y Fj(d)p 1113 1898 74 4 v 1113 1974 a(dt)1196 1918 y Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)1422 1883 y Fr(2)1481 1918 y Fh(\024)c Fs(2)p Fh(f)p Fj(a)1697 1930 y Fr(0)1752 1918 y Fs(+)18 b(\()p Fj(a)1911 1930 y Fr(2)1966 1918 y Fs(+)g Fj(a)2093 1930 y Fr(3)2131 1918 y Fs(\))p Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)2389 1883 y Fr(2)2425 1918 y Fh(g)27 b(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)2720 1883 y Fr(2)2784 1918 y Fj(:)410 b Fs(\(3.1\))523 2122 y(This)28 b(implies)f(that)1321 2265 y(lim)14 b(sup)1370 2330 y Fi(t)p Ff(!1)1589 2265 y Fh(k)p Fj(u)p Fs(\()p Fj(t)p Fs(\))p Fh(k)23 b(\024)f Fj(\032)1968 2277 y Fr(0)2028 2265 y Fs(:=)2139 2153 y Fg(r)p 2222 2153 330 4 v 2347 2208 a Fj(a)2391 2220 y Fr(0)p 2232 2245 310 4 v 2232 2321 a Fh(j)p Fj(a)2299 2333 y Fr(2)2355 2321 y Fs(+)c Fj(a)2482 2333 y Fr(3)2519 2321 y Fh(j)2552 2265 y Fj(;)642 b Fs(\(3.2\))523 2474 y(whic)n(h)26 b(means)f(that)i(the)f(equation)f (has)h(an)g(absorbing)e(ball)i(in)g Fj(L)2618 2444 y Fr(2)2681 2474 y Fs(of)g(radius)f Fj(\032)3065 2486 y Fr(0)3102 2474 y Fs(.)36 b(Let)26 b(us)523 2574 y(denote)i(b)n(y)1376 2706 y Fj(\016)e Fs(=)d(max)o Fh(f)p Fs(0)p Fj(;)14 b Fs(\()1834 2605 y Fg(q)p 1916 2605 256 4 v 1916 2706 a Fj(a)1960 2677 y Fr(2)1960 2728 y(2)2016 2706 y Fs(+)k Fj(b)2135 2677 y Fr(2)2135 2728 y(2)2190 2706 y Fs(+)g(2)p Fj(a)2359 2718 y Fr(2)2396 2706 y Fs(\))p Fh(g)27 b Fj(;)697 b Fs(\(3.3\))523 2871 y(then)28 b(equation)f(\(2)p Fj(:)p Fs(9\))g(implies)777 3039 y(1)p 777 3076 42 4 v 777 3152 a(2)854 3039 y Fj(d)p 839 3076 74 4 v 839 3152 a(dt)922 3095 y Fh(k)p Fj(u)1012 3107 y Fi(x)1053 3095 y Fh(k)1095 3061 y Fr(2)1155 3095 y Fh(\024)22 b Fj(a)1286 3107 y Fr(0)1323 3095 y Fh(k)p Fj(u)1413 3107 y Fi(x)1454 3095 y Fh(k)1496 3061 y Fr(2)1551 3095 y Fh(\000)c Fj(a)1678 3107 y Fr(1)1716 3095 y Fh(k)p Fj(u)1806 3107 y Fi(xx)1884 3095 y Fh(k)1926 3061 y Fr(2)1981 3095 y Fs(+)g Fj(a)2108 3107 y Fr(3)2145 3095 y Fh(k)p Fj(u)p Fh(k)2277 3061 y Fr(2)2313 3095 y Fh(k)p Fj(u)2403 3107 y Fi(x)2444 3095 y Fh(k)2486 3061 y Fr(2)2541 3095 y Fs(+)g Fj(\016)s Fh(k)p Fj(uu)2802 3107 y Fi(x)2843 3095 y Fh(k)2885 3061 y Fr(2)2949 3095 y Fj(:)245 b Fs(\(3.4\))523 3318 y(If)28 b(w)n(e)f(assume)g(that)h Fh(j)p Fj(b)1254 3330 y Fr(2)1291 3318 y Fh(j)23 b(\024)1425 3249 y(p)p 1494 3249 42 4 v 69 x Fs(3)o Fh(j)p Fj(a)1602 3330 y Fr(2)1640 3318 y Fh(j)p Fs(,)k(i.e.)37 b Fj(\016)27 b Fs(=)22 b(0,)27 b(then)i(equation)e(\(3)p Fj(:)p Fs(4\))g(giv)n(es)1076 3486 y(1)p 1076 3523 V 1076 3599 a(2)1152 3486 y Fj(d)p 1137 3523 74 4 v 1137 3599 a(dt)1220 3542 y Fh(k)p Fj(u)1310 3554 y Fi(x)1351 3542 y Fh(k)1393 3508 y Fr(2)1453 3542 y Fh(\024)c Fj(a)1585 3554 y Fr(0)1622 3542 y Fh(k)p Fj(u)1712 3554 y Fi(x)1753 3542 y Fh(k)1795 3508 y Fr(2)1850 3542 y Fh(\000)18 b Fj(a)1977 3554 y Fr(1)2014 3542 y Fh(k)p Fj(u)2104 3554 y Fi(xx)2183 3542 y Fh(k)2225 3508 y Fr(2)2280 3542 y Fs(+)g Fj(a)2407 3554 y Fr(3)2444 3542 y Fh(k)p Fj(u)p Fh(k)2576 3508 y Fr(2)2612 3542 y Fh(k)p Fj(u)2702 3554 y Fi(x)2743 3542 y Fh(k)2785 3508 y Fr(2)2821 3542 y Fj(:)523 3747 y Fs(Observ)n(e)26 b(that)1592 3847 y Fh(k)p Fj(u)1682 3859 y Fi(x)1723 3847 y Fh(k)1765 3813 y Fr(2)1825 3847 y Fh(\024)d(k)p Fj(u)p Fh(k)j(k)p Fj(u)2161 3859 y Fi(xx)2239 3847 y Fh(k)p Fj(;)913 b Fs(\(3.5\))523 3995 y(Th)n(us)27 b(from)h(the)g(ab)r (o)n(v)n(e)e(w)n(e)h(get)1157 4169 y(1)p 1157 4206 42 4 v 1157 4282 a(2)1234 4169 y Fj(d)p 1219 4206 74 4 v 1219 4282 a(dt)1302 4225 y Fh(k)p Fj(u)1392 4237 y Fi(x)1433 4225 y Fh(k)1475 4191 y Fr(2)1535 4225 y Fs(=)22 b(\()p Fj(a)1698 4237 y Fr(0)1754 4225 y Fh(\000)c Fj(a)1881 4237 y Fr(1)1928 4169 y Fh(k)p Fj(u)2018 4181 y Fi(x)2059 4169 y Fh(k)2101 4139 y Fr(2)p 1928 4206 210 4 v 1949 4282 a Fh(k)p Fj(u)p Fh(k)2081 4258 y Fr(2)2166 4225 y Fs(+)g Fj(a)2293 4237 y Fr(3)2330 4225 y Fh(k)p Fj(u)p Fh(k)2462 4191 y Fr(2)2498 4225 y Fs(\))p Fh(k)p Fj(u)2620 4237 y Fi(x)2661 4225 y Fh(k)2703 4191 y Fr(2)2740 4225 y Fj(:)454 b Fs(\(3.6\))523 4469 y(It)39 b(follo)n(ws)e(from)h (equation)f(\(3)p Fj(:)p Fs(2\))h(and)g(equation)g(\(3)p Fj(:)p Fs(6\))g(that)g(for)g Fh(j)p Fj(b)2790 4481 y Fr(2)2827 4469 y Fh(j)i(\024)2996 4400 y(p)p 3065 4400 42 4 v 69 x Fs(3)o Fh(j)p Fj(a)3173 4481 y Fr(2)3211 4469 y Fh(j)e Fs(the)523 4568 y(absorbing)26 b(ball)h(in)h(terms)g(of)f Fh(k)p Fj(u)1576 4580 y Fi(x)1617 4568 y Fh(k)g Fs(has)g(radius)g Fj(\032)2127 4580 y Fr(1)2164 4568 y Fs(,)h(where)1260 4831 y Fj(\032)1303 4843 y Fr(1)1363 4831 y Fs(=)23 b Fj(a)1495 4843 y Fr(0)1532 4686 y Fg(s)p 1615 4686 1013 4 v 145 x Fs(\()1832 4775 y(1)p 1657 4812 391 4 v 1657 4888 a Fj(a)1701 4900 y Fr(1)1738 4888 y Fh(j)p Fj(a)1805 4900 y Fr(2)1861 4888 y Fs(+)18 b Fj(a)1988 4900 y Fr(3)2025 4888 y Fh(j)2058 4831 y Fs(\)\(1)g(+)2390 4775 y Fj(a)2434 4787 y Fr(3)p 2275 4812 310 4 v 2275 4888 a Fh(j)p Fj(a)2342 4900 y Fr(2)2398 4888 y Fs(+)g Fj(a)2525 4900 y Fr(3)2562 4888 y Fh(j)2595 4831 y Fs(\))q Fj(:)566 b Fs(\(3.7\))1934 5147 y(7)p eop %%Page: 8 8 8 7 bop 523 614 a Fs(If)28 b Fh(j)p Fj(b)665 626 y Fr(2)702 614 y Fh(j)23 b Fj(>)836 546 y Fh(p)p 905 546 42 4 v 68 x Fs(3)o Fh(j)p Fj(a)1013 626 y Fr(2)1051 614 y Fh(j)p Fs(,)k(i.e.)37 b Fj(\016)27 b(>)22 b Fs(0,)27 b(then)i(equation)e(\(3)p Fj(:)p Fs(4\))g(giv)n(es)695 782 y(1)p 695 819 V 695 895 a(2)771 782 y Fj(d)p 756 819 74 4 v 756 895 a(dt)839 838 y Fh(k)p Fj(u)929 850 y Fi(x)970 838 y Fh(k)1012 804 y Fr(2)1072 838 y Fs(=)22 b Fj(a)1203 850 y Fr(0)1241 838 y Fh(k)p Fj(u)1331 850 y Fi(x)1371 838 y Fh(k)1413 804 y Fr(2)1469 838 y Fh(\000)c Fj(a)1596 850 y Fr(1)1633 838 y Fh(k)p Fj(u)1723 850 y Fi(xx)1801 838 y Fh(k)1843 804 y Fr(2)1898 838 y Fs(+)g Fj(\016)s Fh(k)p Fj(u)p Fh(k)2153 804 y Fr(2)2153 859 y Fi(L)2199 842 y Fc(1)2262 838 y Fh(k)p Fj(u)2352 850 y Fi(x)2393 838 y Fh(k)2435 804 y Fr(2)2490 838 y Fs(+)g Fj(a)2617 850 y Fr(3)2655 838 y Fh(k)p Fj(u)p Fh(k)2787 804 y Fr(2)2822 838 y Fh(k)p Fj(u)2912 850 y Fi(x)2953 838 y Fh(k)2995 804 y Fr(2)3032 838 y Fj(:)162 b Fs(\(3.8\))523 1047 y(Putting)28 b(\(3)p Fj(:)p Fs(5\))f(in)h(\(3)p Fj(:)p Fs(8\))f(w)n(e)g(ha)n(v)n(e)966 1224 y(1)p 966 1261 42 4 v 966 1337 a(2)1042 1224 y Fj(d)p 1028 1261 74 4 v 1028 1337 a(dt)1111 1280 y Fh(k)p Fj(u)1201 1292 y Fi(x)1241 1280 y Fh(k)1283 1246 y Fr(2)1343 1280 y Fs(=)c(\()p Fj(a)1507 1292 y Fr(0)1563 1280 y Fh(\000)18 b Fj(a)1690 1292 y Fr(1)1737 1224 y Fh(k)p Fj(u)1827 1236 y Fi(x)1868 1224 y Fh(k)1910 1194 y Fr(2)p 1737 1261 210 4 v 1758 1337 a Fh(k)p Fj(u)p Fh(k)1890 1313 y Fr(2)1975 1280 y Fs(+)g Fj(\016)s Fh(k)p Fj(u)p Fh(k)2230 1246 y Fr(2)2230 1301 y Fi(L)2276 1284 y Fc(1)2357 1280 y Fs(+)g Fj(a)2484 1292 y Fr(3)2521 1280 y Fh(k)p Fj(u)p Fh(k)2653 1246 y Fr(2)2689 1280 y Fs(\))p Fh(k)p Fj(u)2811 1292 y Fi(x)2852 1280 y Fh(k)2894 1246 y Fr(2)2931 1280 y Fj(:)263 b Fs(\(3.9\))523 1505 y(It)32 b(is)f(clear)f(that)i(from)f (simple)h(calculus)f(inequalities)g(w)n(e)g(ha)n(v)n(e)f(the)i(follo)n (wing)e(v)n(ersion)523 1604 y(of)e(Agmon's)f(inequalit)n(y:)1447 1785 y Fh(k)p Fj(u)p Fh(k)1579 1751 y Fr(2)1579 1806 y Fi(L)1625 1789 y Fc(1)1711 1785 y Fh(\024)c(k)p Fj(u)p Fh(k)1931 1751 y Fr(2)1985 1785 y Fs(+)18 b(2)p Fh(k)p Fj(u)p Fh(k)26 b(k)p Fj(u)2358 1797 y Fi(x)2399 1785 y Fh(k)p Fj(:)711 b Fs(\(3.10\))523 1966 y(Th)n(us)27 b(it)h(follo)n(ws)f(from)g(\(3)p Fj(:)p Fs(9\))h(and)f(\(3)p Fj(:)p Fs(10\))g(that)640 2143 y(1)p 640 2180 42 4 v 640 2256 a(2)716 2143 y Fj(d)p 701 2180 74 4 v 701 2256 a(dt)784 2199 y Fh(k)p Fj(u)874 2211 y Fi(x)915 2199 y Fh(k)957 2165 y Fr(2)1017 2199 y Fs(=)c(\()p Fj(a)1181 2211 y Fr(0)1236 2199 y Fh(\000)c Fj(a)1364 2211 y Fr(1)1411 2143 y Fh(k)p Fj(u)1501 2155 y Fi(x)1541 2143 y Fh(k)1583 2113 y Fr(2)p 1411 2180 210 4 v 1431 2256 a Fh(k)p Fj(u)p Fh(k)1563 2232 y Fr(2)1649 2199 y Fs(+)f Fj(\016)s Fh(k)p Fj(u)p Fh(k)1904 2165 y Fr(2)1958 2199 y Fs(+)g(2)p Fj(\016)s Fh(k)p Fj(u)p Fh(k)26 b(k)p Fj(u)2371 2211 y Fi(x)2412 2199 y Fh(k)18 b Fs(+)g Fj(a)2599 2211 y Fr(3)2636 2199 y Fh(k)p Fj(u)p Fh(k)2768 2165 y Fr(2)2803 2199 y Fs(\))p Fh(k)p Fj(u)2925 2211 y Fi(x)2967 2199 y Fh(k)3009 2165 y Fr(2)3045 2199 y Fj(:)107 b Fs(\(3.11\))523 2424 y(Again)30 b(after)f(some)g(delicate)h(algebraic)e(manipulation,)j(w)n(e)e(obtain) h(an)f(absorbing)g(ball)523 2523 y(with)f(radius)f Fj(\032)1005 2493 y Ff(0)1005 2544 y Fr(1)1070 2523 y Fs(in)h(terms)f(of)g Fh(k)p Fj(u)1582 2535 y Fi(x)1623 2523 y Fh(k)701 2794 y Fj(\032)744 2759 y Ff(0)744 2814 y Fr(1)805 2794 y Fs(=)1056 2737 y Fj(\016)s(a)1140 2694 y Fr(3)p Fi(=)p Fr(2)1140 2759 y(0)p 902 2774 496 4 v 902 2852 a Fj(a)946 2864 y Fr(1)983 2852 y Fh(j)p Fj(a)1050 2864 y Fr(2)1106 2852 y Fs(+)18 b Fj(a)1233 2864 y Fr(3)1270 2852 y Fh(j)1293 2828 y Fr(3)p Fi(=)p Fr(2)1408 2794 y Fs([1)g(+)1574 2640 y Fg(s)p 1657 2640 1253 4 v 154 x Fs(1)g(+)1810 2737 y Fj(a)1854 2749 y Fr(1)1891 2737 y Fh(j)p Fj(a)1958 2749 y Fr(2)2013 2737 y Fs(+)g Fj(a)2140 2749 y Fr(3)2178 2737 y Fh(j)p Fs(\()p Fh(j)p Fj(a)2300 2749 y Fr(2)2356 2737 y Fs(+)g Fj(a)2483 2749 y Fr(3)2520 2737 y Fh(j)g Fs(+)g Fj(\016)k Fs(+)c Fj(a)2830 2749 y Fr(3)p 1810 2774 1058 4 v 2259 2850 a Fj(\016)2299 2826 y Fr(2)2336 2850 y Fj(a)2380 2862 y Fr(0)2877 2794 y Fs(\))28 b(])p Fj(;)192 b Fs(\(3.12\))523 3050 y(pro)n(vided)36 b(that)i Fh(j)p Fj(b)1123 3062 y Fr(2)1160 3050 y Fh(j)h Fj(>)1326 2982 y Fh(p)p 1395 2982 42 4 v 68 x Fs(3)p Fh(j)p Fj(a)1504 3062 y Fr(2)1541 3050 y Fh(j)p Fs(.)66 b(Applying)38 b(\(3)p Fj(:)p Fs(2\),)i(\(3)p Fj(:)p Fs(7\),)f(and)e(\(3)p Fj(:)p Fs(12\))g(in)h(\(3)p Fj(:)p Fs(10\),)h(w)n(e)523 3150 y(obtain)27 b(an)h(absorbing)e(ball)h(in)h(the)g Fj(L)1734 3120 y Ff(1)1831 3150 y Fs(of)g(radius)f Fj(\032)2219 3162 y Ff(1)2289 3150 y Fs(,)h(where)f(if)h Fh(j)p Fj(b)2715 3162 y Fr(2)2752 3150 y Fh(j)23 b(\024)2886 3081 y(p)p 2955 3081 V 69 x Fs(3)o Fh(j)p Fj(a)3063 3162 y Fr(2)3100 3150 y Fh(j)1102 3411 y Fj(\032)1145 3377 y Fr(2)1145 3432 y Ff(1)1238 3411 y Fs(=)1450 3355 y Fj(a)1494 3367 y Fr(0)p 1336 3392 310 4 v 1336 3468 a Fh(j)p Fj(a)1403 3480 y Fr(2)1459 3468 y Fs(+)18 b Fj(a)1586 3480 y Fr(3)1623 3468 y Fh(j)1674 3411 y Fs(+)1827 3355 y(2)p Fj(a)1913 3312 y Fr(3)p Fi(=)p Fr(2)1913 3377 y(0)p 1767 3392 V 1767 3468 a Fh(j)p Fj(a)1834 3480 y Fr(2)1890 3468 y Fs(+)g Fj(a)2017 3480 y Fr(3)2054 3468 y Fh(j)2087 3266 y Fg(s)p 2170 3266 639 4 v 2200 3355 a Fs(1)p 2180 3392 82 4 v 2180 3468 a Fj(a)2224 3480 y Fr(1)2271 3411 y Fs(\(1)g(+)2571 3355 y Fj(a)2615 3367 y Fr(3)p 2456 3392 310 4 v 2456 3468 a Fh(j)p Fj(a)2523 3480 y Fr(2)2579 3468 y Fs(+)g Fj(a)2706 3480 y Fr(3)2743 3468 y Fh(j)2776 3411 y Fs(\))367 b(\(3.13\))523 3659 y(and)609 3908 y Fj(\032)652 3874 y Fr(2)652 3929 y Ff(1)745 3908 y Fs(=)957 3852 y Fj(a)1001 3864 y Fr(0)p 843 3889 V 843 3965 a Fh(j)p Fj(a)910 3977 y Fr(2)965 3965 y Fs(+)19 b Fj(a)1093 3977 y Fr(3)1130 3965 y Fh(j)1181 3908 y Fs(+)1407 3852 y(2)p Fj(\016)s(a)1533 3822 y Fr(2)1533 3873 y(0)p 1274 3889 429 4 v 1274 3965 a Fj(a)1318 3977 y Fr(1)1355 3965 y Fh(j)p Fj(a)1422 3977 y Fr(2)1478 3965 y Fs(+)f Fj(a)1605 3977 y Fr(3)1642 3965 y Fh(j)1665 3941 y Fr(2)1712 3908 y Fs([1)g(+)1878 3755 y Fg(s)p 1961 3755 1253 4 v 153 x Fs(1)g(+)2114 3852 y Fj(a)2158 3864 y Fr(1)2195 3852 y Fh(j)p Fj(a)2262 3864 y Fr(2)2318 3852 y Fs(+)g Fj(a)2445 3864 y Fr(3)2482 3852 y Fh(j)p Fs(\()p Fh(j)p Fj(a)2604 3864 y Fr(2)2660 3852 y Fs(+)g Fj(a)2787 3864 y Fr(3)2824 3852 y Fh(j)h Fs(+)f Fj(\016)j Fs(+)d Fj(a)3134 3864 y Fr(3)3172 3852 y Fs(\))p 2114 3889 1090 4 v 2580 3965 a Fj(\016)2620 3941 y Fr(2)2657 3965 y Fj(a)2701 3977 y Fr(0)3241 3908 y Fs(])p Fj(;)3175 4071 y Fs(\(3.14\))523 4171 y(otherwise.)39 b(Do)r(ering)28 b(et)h(al.)40 b([1988])27 b(established)i(the)g(sizes)f(of)h(the)g(absorbing)e(balls)h(for)523 4271 y(the)i(cubic)h(Ginzburg-Landau)d(equation)i(\(1.4\),)g(i.e.)45 b(the)30 b(equation)g(\(1)p Fj(:)p Fs(1\))f(without)i(the)523 4370 y(nonlo)r(cal)c(in)n(tegral)f(term.)37 b(They)27 b(obtained)h(the)g(follo)n(wing)e(radii:)1786 4562 y Fj(\032)1829 4574 y Fr(0)1889 4562 y Fh(\030)c Fj(a)2020 4519 y Fr(1)p Fi(=)p Fr(2)2020 4584 y(0)1061 4823 y Fj(\032)1104 4835 y Fr(1)1164 4823 y Fh(\030)1251 4681 y Fg(\()1360 4749 y Fj(a)1404 4761 y Fr(0)2227 4749 y Fs(if)28 b Fh(j)p Fj(b)2362 4761 y Fr(2)2399 4749 y Fh(j)23 b(\024)2533 4680 y(p)p 2602 4680 42 4 v 69 x Fs(3)o Fh(j)p Fj(a)2710 4761 y Fr(2)2748 4749 y Fh(j)1360 4879 y Fj(\016)s(a)1444 4835 y Fr(3)p Fi(=)p Fr(2)1444 4901 y(0)1548 4879 y Fs([1)18 b(+)1714 4785 y Fg(q)p 1797 4785 297 4 v 94 x Fs(1)g(+)1958 4846 y Fr(1+)p Fi(\016)p 1950 4860 134 4 v 1950 4907 a(\016)1982 4891 y Fe(2)2015 4907 y Fi(a)2051 4915 y Fe(0)2093 4879 y Fs(])111 b(if)28 b Fh(j)p Fj(b)2362 4891 y Fr(2)2399 4879 y Fh(j)23 b Fj(>)2533 4810 y Fh(p)p 2602 4810 42 4 v 69 x Fs(3)o Fh(j)p Fj(a)2710 4891 y Fr(2)2748 4879 y Fh(j)1934 5147 y Fs(8)p eop %%Page: 9 9 9 8 bop 523 614 a Fs(and)954 778 y Fj(\032)997 744 y Fr(2)997 799 y Ff(1)1090 778 y Fh(\030)1178 636 y Fg(\()1286 709 y Fj(a)1330 721 y Fr(0)1386 709 y Fs(+)18 b(2)p Fj(a)1555 666 y Fr(3)p Fi(=)p Fr(2)1555 732 y(0)2310 709 y Fs(if)29 b Fh(j)p Fj(b)2446 721 y Fr(2)2482 709 y Fh(j)24 b(\024)2616 641 y(p)p 2685 641 42 4 v 68 x Fs(3)p Fh(j)p Fj(a)2794 721 y Fr(2)2831 709 y Fh(j)1286 839 y Fj(a)1330 851 y Fr(0)1386 839 y Fs(+)18 b(2)p Fj(\016)s(a)1595 809 y Fr(2)1595 860 y(0)1632 839 y Fs([1)g(+)1798 746 y Fg(q)p 1881 746 297 4 v 93 x Fs(1)g(+)2042 806 y Fr(1+)p Fi(\016)p 2034 820 134 4 v 2034 868 a(\016)2066 851 y Fe(2)2098 868 y Fi(a)2134 876 y Fe(0)2177 839 y Fs(])110 b(if)29 b Fh(j)p Fj(b)2446 851 y Fr(2)2482 839 y Fh(j)24 b Fj(>)2616 770 y Fh(p)p 2685 770 42 4 v 69 x Fs(3)p Fh(j)p Fj(a)2794 851 y Fr(2)2831 839 y Fh(j)k Fj(;)523 1012 y Fs(where)37 b Fj(\016)42 b Fs(is)37 b(giv)n(en)h(in)g(\(3)p Fj(:)p Fs(3\).)67 b(Eviden)n(tly)38 b(the)g(sizes)g(of)f(the)i(absorbing)d (balls)i(for)f(the)523 1112 y(nonlo)r(cal)23 b(amplitude)i(equation)e (are)g(di\013eren)n(t)i(from)e(the)i(those)e(of)h(the)h(cubic)f (Ginzburg-)523 1212 y(Landau)j(equation.)36 b(Ho)n(w)n(ev)n(er)25 b(if)j(w)n(e)e(let)i Fj(a)1890 1224 y Fr(1)1950 1212 y Fs(=)23 b(1,)j Fj(a)2173 1224 y Fr(2)2234 1212 y Fs(=)c Fh(\000)p Fs(1)27 b(and)g Fj(a)2660 1224 y Fr(3)2720 1212 y Fs(=)22 b(0)27 b(in)h(\(3)p Fj(:)p Fs(2\),)f(\(3)p Fj(:)p Fs(7\),)523 1311 y(\(3)p Fj(:)p Fs(11\))33 b(\(3)p Fj(:)p Fs(13\)and)g(\(3)p Fj(:)p Fs(14\),)h(then)g(w)n(e)f(ha)n(v)n(e)g (all)g(radii)g(exactly)g(coincide,)i(with)f(those)f(of)523 1411 y(Do)r(ering)27 b(et)h(al.)36 b([1988].)648 1511 y(The)28 b(ab)r(o)n(v)n(e)f(estimates)h(on)g(the)g Fj(L)1737 1480 y Ff(1)1835 1511 y Fs(norm)g(w)n(ere)f(obtained)h(b)n(y)g(using)g (in)n(terp)r(olation)523 1610 y(inequalities.)34 b(Let)20 b(us)g(remark,)g(ho)n(w)n(ev)n(er,)f(that)h(direct)g(and)f(probably)g (sharp)r(er)g(estimates)523 1710 y(migh)n(t)28 b(b)r(e)g(obtained)f(b)n (y)g(implemen)n(ting)h(the)g(tec)n(hniques)f(dev)n(elop)r(ed)h(b)n(y)f (Collet)g([1994].)523 1984 y Fk(4)135 b(Global)45 b(attractor)i(and)d (dimension)i(estimates)523 2166 y Fs(Here)21 b(again)e(w)n(e)i (consider)e(the)j(case)d(when)i Fj(a)1918 2178 y Fr(2)1961 2166 y Fs(+)5 b Fj(a)2075 2178 y Fr(3)2134 2166 y Fj(<)23 b Fs(0)d(and)h Fj(a)2483 2178 y Fr(2)2525 2166 y Fs(+)5 b Fj(ib)2660 2178 y Fr(2)2719 2166 y Fh(6)p Fs(=)23 b(0.)34 b(Let)21 b(us)f(denote)523 2266 y(b)n(y)h Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))24 b(=)e Fj(S)5 b Fs(\()p Fj(t)p Fs(\))p Fj(u)1167 2278 y Fr(0)1204 2266 y Fs(\()p Fj(x)p Fs(\))23 b(the)f(solution)f(op)r(erator.)33 b(Based)20 b(on)h(the)h(Remark)f(at)g(the)h(end)f(of)523 2365 y(section)e(2)f(w)n (e)h(conclude)f(from)h(\(3)p Fj(:)p Fs(2\),)h(\(3)p Fj(:)p Fs(7\))f(and)g(\(3)p Fj(:)p Fs(12\))f(that)h Fj(S)5 b Fs(\()p Fj(t)p Fs(\))19 b(is)g(a)g(b)r(ounded)g(op)r(erator)523 2465 y(from)32 b Fj(L)781 2435 y Fr(2)781 2486 y Fi(per)883 2465 y Fs(\(0)p Fj(;)14 b Fs(1\))32 b(in)n(to)h Fj(H)1350 2435 y Fr(1)1343 2486 y Fi(per)1445 2465 y Fs(\(0)p Fj(;)14 b Fs(1\),)34 b(for)e(ev)n(ery)f Fj(t)h(>)f Fs(0.)51 b(Since)33 b Fj(H)2615 2435 y Fr(1)2608 2486 y Fi(per)2710 2465 y Fs(\(0)p Fj(;)14 b Fs(1\))33 b(is)f(compactly)523 2565 y(im)n(b)r(edded)37 b(in)f Fj(L)1072 2535 y Fr(2)1072 2585 y Fi(per)1174 2565 y Fs(\(0)p Fj(;)14 b Fs(1\),)37 b(then)g Fj(S)5 b Fs(\()p Fj(t)p Fs(\))36 b(is)g(a)f(compact)h(mapping) g(from)f Fj(L)2924 2535 y Fr(2)2924 2585 y Fi(per)3026 2565 y Fs(\(0)p Fj(;)14 b Fs(1\))36 b(in)n(to)523 2664 y Fj(L)580 2634 y Fr(2)580 2685 y Fi(per)682 2664 y Fs(\(0)p Fj(;)14 b Fs(1\))24 b(for)f(ev)n(ery)g Fj(t)g(>)g Fs(0.)35 b(Th)n(us)24 b(the)g(ball)g Fj(B)t Fs(\(0)p Fj(;)14 b(\032)2196 2676 y Fr(0)2233 2664 y Fs(\))23 b(=)g Fh(f)p Fj(w)i Fh(2)f Fj(L)2638 2634 y Fr(2)2674 2664 y Fs(\(0)p Fj(;)14 b Fs(1\))24 b Fh(j)g(k)p Fj(w)r Fh(k)f Fj(<)g(\032)3229 2676 y Fr(0)3266 2664 y Fh(g)g Fs(is)523 2764 y(an)k(absorbing)f(set)i (in)f Fj(L)1300 2734 y Fr(2)1300 2784 y Fi(per)1402 2764 y Fs(\(0)p Fj(;)14 b Fs(1\))27 b(and)h(the)g Fj(!)s Fs(-limit)f(set)h (of)f Fj(B)t Fs(\(0)p Fj(;)14 b(\032)2644 2776 y Fr(0)2681 2764 y Fs(\))28 b(under)g Fj(S)5 b Fs(\()p Fj(t)p Fs(\),)27 b(whic)n(h)523 2864 y(w)n(e)g(denote)h Fh(A)p Fs(,)1400 3075 y Fh(A)c Fs(=)1589 2996 y Fg(\\)1577 3172 y Fi(s>)p Fr(0)1694 3075 y Fs(\()p 1726 2978 563 4 v 1734 2996 a Fg([)1726 3174 y Fi(t)p Ff(\025)p Fi(s)1848 3075 y Fj(S)5 b Fs(\()p Fj(t)p Fs(\))p Fj(B)t Fs(\(0)p Fj(;)14 b(\032)2219 3087 y Fr(0)2256 3075 y Fs(\))q(\))p Fj(;)873 b Fs(\(4.1\))523 3352 y(where)29 b(the)g(closure)f(is)i(tak)n(en)e(in)i (the)f Fj(L)1801 3322 y Fr(2)1867 3352 y Fs(top)r(ology)-7 b(,)29 b(is)g(the)h(global)e(attractor)f(\(see)j(Hale)523 3452 y([1988],)e(T)-7 b(emam)29 b([1988])e(and)i(references)e (therein\).)42 b(The)29 b(global)f(attractor)f(A)i(is)g(neces-)523 3551 y(sarily)d(a)i(nonempt)n(y)f(compact)g(subset)h(of)f Fj(L)1942 3521 y Fr(2)1942 3572 y Fi(per)2044 3551 y Fs(\(0)p Fj(;)14 b Fs(1\).)648 3651 y(W)-7 b(e)26 b(are)f(going)g(to)g (follo)n(w)h(the)g(w)n(orks)e(of)i(Constan)n(tin)g(and)g(F)-7 b(oias)25 b([1988])f(Constan)n(tin)523 3751 y(et)k(al.)36 b([1985],)26 b(Ghidaglia)g(and)i(H)n(\023)-39 b(eron)25 b([1987],)h(Do)r(ering)h(et)h(al.)36 b([1987],)26 b(T)-7 b(emam)27 b([1988],)523 3850 y(Babin)32 b(and)g(Vishik)h([1992])e (\(see)h(also)g(references)f(therein\))i(to)f(obtain)g(the)h(upp)r(er)g (and)523 3950 y(lo)n(w)n(er)c(b)r(ounds)i(of)f(the)h(dimension)g(of)f Fh(A)p Fs(.)46 b(W)-7 b(e)31 b(use)f(the)h(trace)f(form)n(ula)g(of)g (Constan)n(tin-)523 4050 y(F)-7 b(oias-T)g(emam)29 b(to)i(\014nd)g(the) g(dimension)g(of)f(the)h(attractor)e(of)i(the)g(complex)f(Ginzburg-)523 4149 y(Landau)37 b(equation)g(with)h(the)g(extra)f(nonlo)r(cal)f(term.) 67 b(W)-7 b(e)38 b(note)g(that)g(the)g(condition)523 4249 y(under)32 b(whic)n(h)g(the)g(trace)f(form)n(ula)g(of)h(Constan)n (tin-F)-7 b(oias-T)g(emam)30 b(is)i(v)-5 b(alid)32 b(is)g(satis\014ed) 523 4348 y(here)26 b(\(see)g(T)-7 b(emam)27 b([1988]\).)35 b(W)-7 b(e)26 b(no)n(w)g(determine)h(the)g(upp)r(er)f(b)r(ound)h(of)f (the)h(Hausdor\013)523 4448 y(and)c(fractal)f(dimension)h(of)g(the)h (global)e(attractor)g Fh(A)p Fs(.)35 b(In)24 b(the)f(later)g(part)f(of) h(this)h(section,)523 4548 y(w)n(e)j(will)h(also)f(establish)g(a)g(lo)n (w)n(er)f(b)r(ound)i(for)f(the)h(dimension)g(of)f(the)h(global)f (attractor.)648 4647 y(By)38 b(putting)g(the)h(nonlo)r(cal)f(equation)f (\(1)p Fj(:)p Fs(1\))h(in)h(the)f(form)g Fj(u)2663 4659 y Fi(t)2733 4647 y Fs(=)i Fj(F)12 b Fs(\()p Fj(u)p Fs(\))39 b(and)f(then)523 4747 y(linearize)25 b(it)h(around)f(an)n(y)g(\014xed)h (solution)f Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))27 b(in)f(the)g(global) e(attractor)h(w)n(e)g(get)h Fj(U)3271 4759 y Fi(t)3323 4747 y Fs(=)1934 5147 y(9)p eop %%Page: 10 10 10 9 bop 523 614 a Fj(F)588 584 y Ff(0)611 614 y Fs(\()p Fj(u)p Fs(\))p Fj(U)9 b Fs(;)28 b(that)g(is)f Fj(U)37 b Fs(satis\014es)26 b(the)i(ev)n(olution)f(equation:)849 797 y Fj(U)906 809 y Fi(t)1018 797 y Fs(=)82 b Fj(a)1209 809 y Fr(0)1246 797 y Fj(U)27 b Fs(+)18 b(\()p Fj(a)1489 809 y Fr(1)1545 797 y Fs(+)g Fj(ib)1693 809 y Fr(1)1730 797 y Fs(\))p Fj(U)1819 809 y Fi(xx)1917 797 y Fs(+)g(\()p Fj(a)2076 809 y Fr(2)2132 797 y Fs(+)g Fj(ib)2280 809 y Fr(2)2316 797 y Fs(\)[)p Fh(j)p Fj(u)p Fh(j)2465 763 y Fr(2)2502 797 y Fj(U)27 b Fs(+)19 b(2)p Fj(uR)q(e)p Fs(\()5 b(\026)-47 b Fj(u)n(U)9 b Fs(\)])1165 981 y(+\()p Fj(a)1306 993 y Fr(3)1362 981 y Fs(+)18 b Fj(ib)1510 993 y Fr(3)1546 981 y Fs(\)[2)p Fj(u)1705 868 y Fg(Z)1787 888 y Fr(1)1750 1057 y(0)1838 981 y Fj(R)q(e)p Fs(\()p Fj(u)2035 960 y Fs(\026)2021 981 y Fj(U)8 b Fs(\))p Fj(dx)20 b Fs(+)e Fj(U)2390 868 y Fg(Z)2473 888 y Fr(1)2436 1057 y(0)2524 981 y Fh(j)p Fj(u)p Fh(j)2618 947 y Fr(2)2655 981 y Fj(dx)p Fs(])449 b(\(4.2\))523 1215 y(with)37 b Fj(U)9 b Fs(\(0\))37 b(=)g Fj(\030)k Fh(2)c Fj(L)1258 1185 y Fr(2)1258 1236 y Fi(per)1360 1215 y Fs(\(0)p Fj(;)14 b Fs(1\))36 b(and)g Fj(U)9 b Fs(\()p Fj(x;)14 b(t)p Fs(\))37 b(satis\014es)e(the)i(same)e(p)r(erio)r(dic)h(b)r(oundary)523 1315 y(conditions)d(as)g Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\).)54 b(Let)34 b Fj(\030)i Fs(=)d Fj(\030)1698 1327 y Fr(1)1735 1315 y Fj(;)14 b(:)g(:)g(:)g(;)g(\030)1956 1327 y Fi(m)2052 1315 y Fs(b)r(e)34 b Fj(m)f Fs(linearly)g(indep)r (enden)n(t)h(functions)523 1415 y(of)e Fj(L)679 1385 y Fr(2)679 1435 y Fi(per)781 1415 y Fs(\(0)p Fj(;)14 b Fs(1\).)48 b(W)-7 b(e)32 b(solv)n(e)e(\(4.2\))i(with)g(initial)g (conditions)f Fj(U)9 b Fs(\()p Fj(x;)14 b Fs(0\))30 b(=)f Fj(\030)2848 1427 y Fi(i)2876 1415 y Fj(;)14 b(i)29 b Fs(=)g(1)p Fj(;)14 b Fh(\001)g(\001)g(\001)f Fj(;)h(m)p Fs(,)523 1514 y(and)40 b(denote)g(the)g(corresp)r(onding)e(solutions)i (b)n(y)f Fj(U)53 b Fs(=)43 b Fj(U)2440 1526 y Fr(1)2477 1514 y Fj(;)14 b(:)g(:)g(:)g(;)g(U)2719 1526 y Fi(m)2782 1514 y Fs(.)74 b(W)-7 b(e)40 b(set)g Fj(P)3229 1526 y Fi(m)3293 1514 y Fs(\()p Fj(t)p Fs(\))523 1614 y(to)k(b)r(e)g(the)h (orthonogonal)c(pro)5 b(jection)43 b(of)h Fj(L)2028 1584 y Fr(2)2028 1634 y Fi(per)2130 1614 y Fs(\(0)p Fj(;)14 b Fs(1\))44 b(on)n(to)f(the)i(space)e(spanned)h(b)n(y)523 1714 y Fh(f)p Fj(U)622 1726 y Fr(1)658 1714 y Fs(\()p Fj(t)p Fs(\))p Fj(;)14 b(:)g(:)g(:)h(;)f(U)995 1726 y Fi(m)1057 1714 y Fs(\()p Fj(t)p Fs(\))p Fh(g)p Fs(.)60 b(A)n(t)35 b(an)n(y)g(giv)n(en)f(time)i Fj(\034)9 b Fs(,)38 b(let)d Fh(f)p Fj(\036)2313 1726 y Fi(j)2348 1714 y Fs(\()p Fj(\034)9 b Fs(\))p Fh(g)2499 1683 y Fi(m)2499 1735 y(j)s Fr(=1)2654 1714 y Fs(b)r(e)36 b(a)f(subset)g(of)g Fj(H)3292 1683 y Fr(1)3285 1734 y Fi(per)523 1822 y Fs(whic)n(h)26 b(is)g(an)g(orthonormal)e(basis,)i(with)h(resp)r(ect)e(to)i(the)f Fj(L)2424 1792 y Fr(2)2424 1843 y Fi(per)2526 1822 y Fs(\(0)p Fj(;)14 b Fs(1\))26 b(inner)g(pro)r(duct,)g(i.e.)523 1922 y(\()p Fj(\036)604 1934 y Fi(i)632 1922 y Fs(\()p Fj(\034)9 b Fs(\))p Fj(;)14 b(\036)827 1934 y Fi(j)864 1922 y Fs(\()p Fj(\034)9 b Fs(\)\))25 b(=)d Fj(\016)1154 1934 y Fi(ij)1213 1922 y Fs(,)27 b(of)h(the)g(linear)f(space)g Fj(P)2007 1934 y Fi(m)2070 1922 y Fs(\()p Fj(\034)9 b Fs(\))p Fj(L)2236 1892 y Fr(2)2236 1943 y Fi(per)2339 1922 y Fs(,)28 b(i.e.)1027 2105 y(span)p Fh(f)p Fj(\036)1285 2117 y Fr(1)1322 2105 y Fs(\()p Fj(\034)9 b Fs(\))p Fj(;)14 b Fh(\001)g(\001)g(\001)h Fj(;)f(\036)1666 2117 y Fi(m)1729 2105 y Fs(\()p Fj(\034)9 b Fs(\))p Fh(g)24 b Fs(=)f(span)o Fh(f)p Fj(U)2257 2117 y Fr(1)2294 2105 y Fs(\()p Fj(\034)9 b Fs(\))p Fj(;)14 b(:)g(:)g(:)h(;)f(U)2646 2117 y Fi(m)2708 2105 y Fs(\()p Fj(\034)9 b Fs(\))p Fh(g)p Fj(:)523 2287 y Fs(Note)28 b(that)912 2516 y Fj(R)q(e)p Fh(f)p Fs(T)-7 b(rac)n(\()p Fj(F)1317 2481 y Ff(0)1341 2516 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(P)1600 2528 y Fi(m)1664 2516 y Fs(\()p Fj(t)p Fs(\)\))p Fh(g)23 b Fs(=)1973 2412 y Fi(m)1943 2437 y Fg(X)1945 2614 y Fi(j)s Fr(=1)2077 2516 y Fj(R)q(e)p Fs(\()p Fj(F)2277 2481 y Ff(0)2299 2516 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)2554 2528 y Fi(j)2591 2516 y Fs(\()p Fj(t)p Fs(\))p Fj(;)14 b(\036)2771 2528 y Fi(j)2806 2516 y Fs(\()p Fj(t)p Fs(\)\))29 b Fj(;)233 b Fs(\(4.3\))523 2782 y(and)1065 2992 y Fj(R)q(e)p Fs(\()p Fj(F)1265 2958 y Ff(0)1287 2992 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1542 3004 y Fi(j)1578 2992 y Fs(\()p Fj(t)p Fs(\))p Fj(;)14 b(\036)1758 3004 y Fi(j)1794 2992 y Fs(\()p Fj(t)p Fs(\)\))24 b(=)f Fj(a)2076 3004 y Fr(0)2113 2992 y Fh(k)p Fj(\036)2204 3004 y Fi(j)2239 2992 y Fs(\()p Fj(t)p Fs(\))p Fh(k)2375 2958 y Fr(2)2431 2992 y Fh(\000)18 b Fj(a)2558 3004 y Fr(1)2595 2992 y Fh(k)2647 2936 y Fj(@)5 b(\036)2745 2948 y Fi(j)2780 2936 y Fs(\()p Fj(t)p Fs(\))p 2647 2973 228 4 v 2712 3049 a Fj(@)g(x)2884 2992 y Fh(k)2926 2958 y Fr(2)2963 2992 y Fs(+)983 3212 y Fj(R)q(e)p Fh(f)p Fs(\()p Fj(a)1204 3224 y Fr(2)1259 3212 y Fs(+)18 b Fj(ib)1407 3224 y Fr(2)1443 3212 y Fs(\))1489 3099 y Fg(Z)1573 3120 y Fr(1)1536 3288 y(0)1610 3212 y Fs(\()p Fh(j)p Fj(u)p Fh(j)1736 3178 y Fr(2)1773 3212 y Fh(j)p Fj(\036)1845 3224 y Fi(j)1881 3212 y Fs(\()p Fj(t)p Fs(\))p Fh(j)1998 3178 y Fr(2)2054 3212 y Fs(+)g(2)p Fj(u)2238 3190 y Fs(\026)2227 3212 y Fj(\036)2276 3224 y Fi(j)2310 3212 y Fs(\()p Fj(t)p Fs(\))q Fj(R)q(e)p Fs(\()5 b(\026)-47 b Fj(u)o(\036)2636 3224 y Fi(j)2671 3212 y Fs(\()p Fj(t)p Fs(\)\)\))p Fj(dx)p Fh(g)p Fs(+)717 3451 y Fj(R)q(e)p Fh(f)p Fs(\()p Fj(a)938 3463 y Fr(3)993 3451 y Fs(+)18 b Fj(ib)1141 3463 y Fr(3)1177 3451 y Fs(\)[2\()p Fj(R)q(e)1423 3338 y Fg(Z)1505 3358 y Fr(1)1469 3527 y(0)1557 3451 y Fj(u\036)1654 3463 y Fi(j)1689 3451 y Fs(\()p Fj(t)p Fs(\))p Fj(dx)p Fs(\))1919 3338 y Fg(Z)2003 3358 y Fr(1)1966 3527 y(0)2054 3451 y Fj(u)2113 3429 y Fs(\026)2102 3451 y Fj(\036)2151 3463 y Fi(j)2186 3451 y Fs(\()p Fj(t)p Fs(\))q Fj(dx)h Fs(+)f Fh(k)p Fj(u)p Fh(k)2605 3417 y Fr(2)2640 3451 y Fh(k)p Fj(\036)2731 3463 y Fi(j)2766 3451 y Fs(\()p Fj(t)p Fs(\))p Fh(k)2902 3417 y Fr(2)2940 3451 y Fs(])p Fh(g)p Fj(:)189 b Fs(\(4.4\))523 3675 y(But)1145 3774 y Fh(j)p Fj(u)p Fh(j)1239 3740 y Fr(2)1276 3774 y Fh(j)p Fj(\036)1348 3786 y Fi(j)1384 3774 y Fh(j)1407 3740 y Fr(2)1463 3774 y Fs(+)18 b(2)p Fj(u)1647 3752 y Fs(\026)1636 3774 y Fj(\036)1685 3786 y Fi(j)1719 3774 y Fj(R)q(e)p Fs(\()5 b(\026)-47 b Fj(u)o(\036)1950 3786 y Fi(j)1986 3774 y Fs(\))23 b(=)g(2)p Fh(j)p Fj(u)p Fh(j)2265 3740 y Fr(2)2301 3774 y Fh(j)p Fj(\036)2373 3786 y Fi(j)2409 3774 y Fh(j)2432 3740 y Fr(2)2487 3774 y Fs(+)18 b Fj(u)2618 3740 y Fr(2)2666 3752 y Fs(\026)2655 3774 y Fj(\036)2704 3740 y Fr(2)2704 3795 y Fi(j)2742 3774 y Fj(;)523 3924 y Fs(th)n(us)28 b(agrue)e(similarly)h(to)g(\(3)p Fj(:)p Fs(4\))g(w)n(e)h(obtain)565 4166 y Fj(R)q(e)p Fh(f)p Fs(\()p Fj(a)786 4178 y Fr(2)826 4166 y Fs(+)t Fj(ib)960 4178 y Fr(2)996 4166 y Fs(\))1042 4053 y Fg(Z)1125 4073 y Fr(1)1088 4241 y(0)1163 4166 y Fs(\()p Fh(j)p Fj(u)p Fh(j)1289 4131 y Fr(2)1326 4166 y Fh(j)p Fj(\036)1398 4178 y Fi(j)1433 4166 y Fs(\()p Fj(t)p Fs(\))p Fh(j)1550 4131 y Fr(2)1592 4166 y Fs(+)t(2)p Fj(u)1762 4144 y Fs(\026)1751 4166 y Fj(\036)1800 4178 y Fi(j)1835 4166 y Fs(\()p Fj(t)p Fs(\))p Fj(R)q(e)p Fs(\()5 b(\026)-47 b Fj(u)o(\036)2160 4178 y Fi(j)2196 4166 y Fs(\()p Fj(t)p Fs(\)\)\))p Fj(dx)p Fh(g)24 b(\024)f Fj(\016)2652 4053 y Fg(Z)2735 4073 y Fr(1)2698 4241 y(0)2786 4166 y Fh(j)p Fj(u)p Fh(j)2880 4131 y Fr(2)2917 4166 y Fh(j)p Fj(\036)2989 4178 y Fi(j)3024 4166 y Fh(j)3047 4131 y Fr(2)3084 4166 y Fj(dx)43 b Fs(\(4.5\))523 4395 y(where)27 b Fj(\016)k Fs(is)c(giv)n(en)g(b)n(y)g(\(3)p Fj(:)p Fs(3\).)37 b(Also)1013 4637 y Fj(R)q(e)p Fh(f)p Fs(\()p Fj(a)1234 4649 y Fr(3)1288 4637 y Fs(+)18 b Fj(ib)1436 4649 y Fr(3)1473 4637 y Fs(\)[2\()p Fj(R)q(e)1719 4524 y Fg(Z)1801 4544 y Fr(1)1764 4713 y(0)1852 4637 y Fj(u\036)1949 4649 y Fi(j)1984 4637 y Fs(\()p Fj(t)p Fs(\))p Fj(dx)p Fs(\))2214 4524 y Fg(Z)2299 4544 y Fr(1)2262 4713 y(0)2350 4637 y Fj(u)2409 4615 y Fs(\026)2398 4637 y Fj(\036)2447 4649 y Fi(j)2482 4637 y Fs(\()p Fj(t)p Fs(\))p Fj(dx)p Fs(+)1337 4854 y Fh(k)p Fj(u)p Fh(k)1469 4819 y Fr(2)1504 4854 y Fh(k)p Fj(\036)1595 4866 y Fi(j)1630 4854 y Fs(\()p Fj(t)p Fs(\))p Fh(k)1766 4819 y Fr(2)1804 4854 y Fs(])p Fh(g)k(\024)h Fs(3)2021 4752 y Fg(q)p 2104 4752 256 4 v 102 x Fj(a)2148 4825 y Fr(2)2148 4876 y(3)2203 4854 y Fs(+)18 b Fj(b)2322 4825 y Fr(2)2322 4876 y(3)2359 4854 y Fh(k)p Fj(u)p Fh(k)2491 4819 y Fr(2)2527 4854 y Fh(k)p Fj(\036)2618 4866 y Fi(j)2653 4854 y Fh(k)2695 4819 y Fr(2)3217 4854 y Fs(\(4.6\))1914 5147 y(10)p eop %%Page: 11 11 11 10 bop 523 614 a Fs(Inserting)31 b(equations)f(\(4)p Fj(:)p Fs(5\))h(and)g(\(4)p Fj(:)p Fs(6\))g(in)h(\(4)p Fj(:)p Fs(4\))f(and)g(using)g(the)g(fact)h(that)f Fh(f)p Fj(\036)3097 626 y Fi(j)3132 614 y Fs(\()p Fj(t)p Fs(\))p Fh(g)3268 584 y Ff(1)3268 636 y Fi(j)s Fr(=1)523 723 y Fs(is)d(an)f(orthonormal)e(basis)i(in)h Fj(L)1555 693 y Fr(2)1555 744 y Fi(per)1657 723 y Fs(\(0)p Fj(;)14 b Fs(1\))27 b(at)h(an)n(y)f(giv)n(en)f(time)i Fj(t)g Fs(w)n(e)f(obtain:)527 960 y Fj(R)q(e)p Fs(\()p Fj(F)727 926 y Ff(0)750 960 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1005 972 y Fi(j)1041 960 y Fs(\()p Fj(t)p Fs(\))p Fj(;)14 b(\036)1221 972 y Fi(j)1257 960 y Fs(\()p Fj(t)p Fs(\)\))24 b Fh(\024)f Fj(a)1539 972 y Fr(0)1594 960 y Fh(\000)18 b Fj(a)1721 972 y Fr(1)1758 960 y Fh(k)1810 904 y Fj(@)5 b(\036)1908 916 y Fi(j)p 1810 941 134 4 v 1828 1017 a Fj(@)g(x)1953 960 y Fh(k)1995 926 y Fr(2)2050 960 y Fs(+)18 b Fj(\016)2187 847 y Fg(Z)2270 868 y Fr(1)2233 1036 y(0)2321 960 y Fh(j)p Fj(u)p Fh(j)2415 926 y Fr(2)2452 960 y Fh(j)p Fj(\036)2524 972 y Fi(j)2560 960 y Fh(j)2583 926 y Fr(2)2620 960 y Fj(dx)h Fs(+)f(3)2854 859 y Fg(q)p 2937 859 256 4 v 101 x Fj(a)2981 931 y Fr(2)2981 982 y(3)3036 960 y Fs(+)g Fj(b)3155 931 y Fr(2)3155 982 y(3)3192 960 y Fh(k)p Fj(u)p Fh(k)3324 926 y Fr(2)3360 960 y Fj(:)523 1178 y Fs(Let)28 b(us)f(denote)h(b)n(y)f Fj(f)32 b Fs(=)1321 1116 y Fg(P)1409 1137 y Fi(m)1409 1203 y(j)s Fr(=1)1542 1178 y Fh(j)p Fj(\036)1614 1190 y Fi(j)1649 1178 y Fh(j)1672 1148 y Fr(2)1710 1178 y Fs(.)37 b(It)28 b(follo)n(ws)e(that)632 1323 y Fi(m)601 1348 y Fg(X)604 1525 y Fi(j)s Fr(=1)735 1427 y Fj(R)q(e)p Fs(\()p Fj(F)935 1393 y Ff(0)958 1427 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1213 1439 y Fi(j)1249 1427 y Fj(;)14 b(\036)1335 1439 y Fi(j)1370 1427 y Fs(\))84 b Fh(\024)e Fj(ma)1750 1439 y Fr(0)1806 1427 y Fh(\000)18 b Fj(a)1933 1439 y Fr(1)2014 1323 y Fi(m)1984 1348 y Fg(X)1986 1525 y Fi(j)s Fr(=1)2117 1427 y Fh(k)2169 1371 y Fj(@)5 b(\036)2267 1383 y Fi(j)p 2169 1408 134 4 v 2187 1484 a Fj(@)g(x)2312 1427 y Fh(k)2354 1393 y Fr(2)2409 1427 y Fs(+)18 b Fj(\016)2546 1314 y Fg(Z)2629 1334 y Fr(1)2592 1502 y(0)2680 1427 y Fh(j)p Fj(u)p Fs(\()p Fj(x;)c(t)p Fs(\))p Fh(j)2952 1393 y Fr(2)2990 1427 y Fj(f)9 b Fs(\()p Fj(x;)14 b(t)p Fs(\))p Fj(dx)1633 1685 y Fs(+3)p Fj(m)1813 1584 y Fg(q)p 1895 1584 256 4 v 1895 1685 a Fj(a)1939 1657 y Fr(2)1939 1707 y(3)1995 1685 y Fs(+)k Fj(b)2114 1657 y Fr(2)2114 1707 y(3)2178 1685 y Fh(k)p Fj(u)p Fh(k)2310 1651 y Fr(2)2346 1685 y Fj(:)848 b Fs(\(4.7\))523 1868 y(By)33 b(Sob)r(olev-Lieb-Thirring)f (inequalit)n(y)-7 b(,)35 b(\(see)e(T)-7 b(emam)34 b(1988)d(p.466\))i (there)h(exist)f(t)n(w)n(o)523 1967 y(absolute)28 b(constan)n(ts)f Fj(c)1256 1979 y Fr(1)1322 1967 y Fs(and)h Fj(c)1520 1979 y Fr(2)1558 1967 y Fs(,)g(whic)n(h)h(are)e(indep)r(enden)n(t)j(of) e Fj(m)g Fs(and)h(of)f(the)h(functions)523 2067 y Fh(f)p Fj(\036)614 2079 y Fi(j)649 2067 y Fh(g)691 2037 y Fi(m)691 2089 y(j)s Fr(=1)810 2067 y Fs(,)e(suc)n(h)h(that)1178 2312 y Fh(k)p Fj(f)9 b Fh(k)1312 2278 y Fr(3)1312 2335 y Fi(L)1358 2318 y Fe(3)1388 2335 y Fr(\(0)p Fi(;)p Fr(1\))1553 2312 y Fh(\024)23 b Fj(c)1677 2324 y Fr(1)1728 2199 y Fg(Z)1811 2220 y Fr(1)1774 2388 y(0)1862 2312 y Fj(f)9 b Fs(\()p Fj(x)p Fs(\))p Fj(dx)20 b Fs(+)e Fj(c)2252 2324 y Fr(2)2333 2208 y Fi(m)2303 2233 y Fg(X)2305 2410 y Fi(j)s Fr(=1)2436 2312 y Fh(k)2488 2256 y Fj(@)5 b(\036)2586 2268 y Fi(j)p 2488 2293 134 4 v 2506 2369 a Fj(@)g(x)2631 2312 y Fh(k)2673 2278 y Fr(2)2710 2312 y Fj(:)523 2567 y Fs(Using)27 b(the)h(fact)g(that)g Fh(k)p Fj(\036)1334 2579 y Fi(j)1369 2567 y Fh(k)23 b Fs(=)f(1,)27 b(it)h(follo)n(ws)f (that)1116 2690 y Fg(Z)1199 2711 y Fr(1)1162 2879 y(0)1250 2803 y Fj(f)9 b Fs(\()p Fj(x)p Fs(\))p Fj(dx)24 b Fs(=)1643 2699 y Fi(m)1613 2724 y Fg(X)1615 2901 y Fi(j)s Fr(=1)1746 2690 y Fg(Z)1829 2711 y Fr(1)1793 2879 y(0)1880 2803 y Fh(j)p Fj(\036)1952 2815 y Fi(j)1988 2803 y Fh(j)2011 2769 y Fr(2)2048 2803 y Fj(dx)g Fs(=)2280 2699 y Fi(m)2250 2724 y Fg(X)2252 2901 y Fi(j)s Fr(=1)2383 2803 y Fh(k)p Fj(\036)2474 2815 y Fi(j)2509 2803 y Fh(k)2551 2769 y Fr(2)2611 2803 y Fs(=)f Fj(m;)523 3053 y Fs(and)k(so)g(w)n(e)h(ha)n(v)n (e)1341 3199 y Fh(k)p Fj(f)9 b Fh(k)1475 3164 y Fr(3)1475 3221 y Fi(L)1521 3205 y Fe(3)1552 3221 y Fr(\(0)p Fi(;)p Fr(1\))1717 3199 y Fh(\024)22 b Fj(c)1840 3211 y Fr(1)1878 3199 y Fj(m)c Fs(+)g Fj(c)2088 3211 y Fr(2)2169 3095 y Fi(m)2139 3120 y Fg(X)2141 3297 y Fi(j)s Fr(=1)2273 3199 y Fh(k)2325 3143 y Fj(@)5 b(\036)2423 3155 y Fi(j)p 2324 3180 V 2343 3256 a Fj(@)g(x)2467 3199 y Fh(k)2509 3164 y Fr(2)2546 3199 y Fj(:)648 b Fs(\(4.8\))523 3427 y(Substitute)29 b(\(4)p Fj(:)p Fs(8\))e(in)h(\(4)p Fj(:)p Fs(7\))f(to)h(obtain)886 3556 y Fi(m)855 3581 y Fg(X)858 3758 y Fi(j)s Fr(=1)989 3660 y Fj(R)q(e)p Fs(\()p Fj(F)1189 3626 y Ff(0)1212 3660 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1467 3672 y Fi(j)1503 3660 y Fj(;)14 b(\036)1589 3672 y Fi(j)1624 3660 y Fs(\))23 b Fh(\024)g Fj(ma)1884 3672 y Fr(0)1940 3660 y Fs(+)2033 3604 y Fj(a)2077 3616 y Fr(1)2114 3604 y Fj(c)2150 3616 y Fr(1)2187 3604 y Fj(m)p 2033 3641 228 4 v 2110 3717 a(c)2146 3729 y Fr(2)2288 3660 y Fh(\000)2381 3604 y Fj(a)2425 3616 y Fr(1)p 2381 3641 82 4 v 2385 3717 a Fj(c)2421 3729 y Fr(2)2472 3660 y Fh(k)p Fj(f)9 b Fh(k)2606 3626 y Fr(3)2606 3683 y Fi(L)2652 3666 y Fe(3)2683 3683 y Fr(\(0)p Fi(;)p Fr(1\))2825 3660 y Fs(+)1657 3941 y Fj(\016)1711 3828 y Fg(Z)1794 3848 y Fr(1)1757 4016 y(0)1845 3941 y Fh(j)p Fj(u)p Fh(j)1939 3906 y Fr(2)1976 3941 y Fj(f)g(dx)19 b Fs(+)f(3)p Fj(m)2333 3839 y Fg(q)p 2415 3839 256 4 v 2415 3941 a Fj(a)2459 3912 y Fr(2)2459 3963 y(3)2515 3941 y Fs(+)g Fj(b)2634 3912 y Fr(2)2634 3963 y(3)2698 3941 y Fh(k)p Fj(u)p Fh(k)2830 3906 y Fr(2)2866 3941 y Fj(:)328 b Fs(\(4.9\))523 4154 y(By)27 b(using)h(H\177)-42 b(older)27 b(inequalit)n(y)-7 b(,)27 b(with)h Fj(p)23 b Fs(=)f(3)p Fj(=)p Fs(2)27 b(and)g Fj(q)f Fs(=)d(3,)k(w)n(e)g(get)1108 4379 y Fj(\016)1162 4266 y Fg(Z)1245 4287 y Fr(1)1208 4455 y(0)1296 4379 y Fh(j)p Fj(u)p Fs(\()p Fj(x;)14 b(t)p Fs(\))p Fh(j)1568 4345 y Fr(2)1606 4379 y Fj(f)9 b Fs(\()p Fj(x;)14 b(t)p Fs(\))p Fj(dx)24 b Fh(\024)f Fj(\016)s Fh(k)p Fj(u)p Fh(k)2208 4345 y Fr(2)2208 4402 y Fi(L)2254 4385 y Fe(3)2284 4402 y Fr(\(0)p Fi(;)p Fr(1\))2426 4379 y Fh(k)p Fj(f)9 b Fh(k)2560 4394 y Fi(L)2606 4378 y Fe(3)2637 4394 y Fr(\(0)p Fi(;)p Fr(1\))2779 4379 y Fj(:)523 4588 y Fs(W)-7 b(e)28 b(then)g(apply)f(Y)-7 b(oung's)28 b(inequalit)n(y)f (to)g(the)h(righ)n(t)f(hand)h(side)f(to)h(get)1042 4839 y Fj(\016)1096 4726 y Fg(Z)1179 4746 y Fr(1)1142 4915 y(0)1230 4839 y Fh(j)p Fj(u)p Fh(j)1324 4805 y Fr(2)1361 4839 y Fj(f)9 b(dx)23 b Fh(\024)1638 4783 y Fj(a)1682 4795 y Fr(1)p 1622 4820 115 4 v 1622 4896 a Fs(2)p Fj(c)1700 4908 y Fr(2)1746 4839 y Fh(k)p Fj(f)9 b Fh(k)1880 4805 y Fr(3)1880 4861 y Fi(L)1926 4845 y Fe(3)1957 4861 y Fr(\(0)p Fi(;)p Fr(1\))2117 4839 y Fs(+)2200 4686 y Fg(s)p 2283 4686 212 4 v 2293 4783 a Fs(8)p Fj(\016)2375 4759 y Fr(3)2412 4783 y Fj(c)2448 4795 y Fr(2)p 2293 4820 192 4 v 2307 4896 a Fs(27)p Fj(a)2435 4908 y Fr(1)2495 4839 y Fh(k)p Fj(u)p Fh(k)2627 4805 y Fr(3)2627 4861 y Fi(L)2673 4845 y Fe(3)2704 4861 y Fr(\(0)p Fi(;)p Fr(1\))2846 4839 y Fj(:)306 b Fs(\(4.10\))1914 5147 y(11)p eop %%Page: 12 12 12 11 bop 523 614 a Fs(Putting)28 b(\(4)p Fj(:)p Fs(10\))e(in)i(\(4)p Fj(:)p Fs(9\))g(w)n(e)f(ha)n(v)n(e)800 760 y Fi(m)769 785 y Fg(X)772 961 y Fi(j)s Fr(=1)903 864 y Fj(R)q(e)p Fs(\()p Fj(F)1103 829 y Ff(0)1126 864 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1381 876 y Fi(j)1417 864 y Fj(;)14 b(\036)1503 876 y Fi(j)1538 864 y Fs(\))84 b Fh(\024)e Fs(\()p Fj(a)1877 876 y Fr(0)1933 864 y Fs(+)2026 807 y Fj(a)2070 819 y Fr(1)2107 807 y Fj(c)2143 819 y Fr(1)p 2026 844 155 4 v 2067 920 a Fj(c)2103 932 y Fr(2)2209 864 y Fs(+)18 b(3)2334 762 y Fg(q)p 2416 762 256 4 v 2416 864 a Fj(a)2460 835 y Fr(2)2460 886 y(3)2516 864 y Fs(+)g Fj(b)2635 835 y Fr(2)2635 886 y(3)2672 864 y Fh(k)p Fj(u)p Fh(k)2804 829 y Fr(2)2840 864 y Fs(\))p Fj(m)g Fh(\000)1828 1118 y Fj(a)1872 1130 y Fr(1)p 1811 1155 115 4 v 1811 1231 a Fs(2)p Fj(c)1889 1243 y Fr(2)1936 1174 y Fh(k)p Fj(f)9 b Fh(k)2070 1140 y Fr(3)2070 1197 y Fi(L)2116 1180 y Fe(3)2146 1197 y Fr(\(0)p Fi(;)p Fr(1\))2307 1174 y Fs(+)2390 1021 y Fg(s)p 2473 1021 212 4 v 2483 1118 a Fs(8)p Fj(\016)2565 1094 y Fr(3)2601 1118 y Fj(c)2637 1130 y Fr(2)p 2483 1155 192 4 v 2497 1231 a Fs(27)p Fj(a)2625 1243 y Fr(1)2685 1174 y Fh(k)p Fj(u)p Fh(k)2817 1140 y Fr(3)2817 1197 y Fi(L)2863 1180 y Fe(3)2893 1197 y Fr(\(0)p Fi(;)p Fr(1\))3035 1174 y Fj(:)117 b Fs(\(4.11\))523 1416 y(Note)28 b(that)g(since)1258 1561 y Fj(m)23 b Fs(=)1473 1457 y Fi(m)1442 1482 y Fg(X)1445 1659 y Fi(j)s Fr(=1)1576 1561 y Fh(k)p Fj(\036)1667 1573 y Fi(j)1702 1561 y Fh(k)1744 1527 y Fr(2)1804 1561 y Fs(=)1891 1448 y Fg(Z)1974 1469 y Fr(1)1937 1637 y(0)2025 1561 y Fj(f)9 b(dx)24 b Fh(\024)e(k)p Fj(f)9 b Fh(k)2410 1576 y Fi(L)2456 1559 y Fe(3)2487 1576 y Fr(\(0)p Fi(;)p Fr(1\))2629 1561 y Fj(;)523 1799 y Fs(so)27 b(that)h(\(4)p Fj(:)p Fs(11\))f(b)r(ecomes)843 1944 y Fi(m)812 1969 y Fg(X)815 2146 y Fi(j)s Fr(=1)946 2048 y Fj(R)q(e)p Fs(\()p Fj(F)1146 2014 y Ff(0)1169 2048 y Fs(\()p Fj(u)p Fs(\()p Fj(s)p Fs(\)\))p Fj(\036)1433 2060 y Fi(j)1469 2048 y Fj(;)14 b(\036)1555 2060 y Fi(j)1590 2048 y Fs(\))83 b Fh(\024)g Fs(\()p Fj(a)1929 2060 y Fr(0)1985 2048 y Fs(+)2078 1992 y Fj(a)2122 2004 y Fr(1)2159 1992 y Fj(c)2195 2004 y Fr(1)p 2078 2029 155 4 v 2119 2105 a Fj(c)2155 2117 y Fr(2)2261 2048 y Fs(+)18 b(3)2386 1947 y Fg(q)p 2468 1947 256 4 v 2468 2048 a Fj(a)2512 2020 y Fr(2)2512 2070 y(3)2568 2048 y Fs(+)g Fj(b)2687 2020 y Fr(2)2687 2070 y(3)2724 2048 y Fh(k)p Fj(u)p Fh(k)2856 2014 y Fr(2)2891 2048 y Fs(\))p Fj(m)h Fh(\000)1863 2303 y Fj(a)1907 2315 y Fr(1)1944 2303 y Fj(m)2017 2272 y Fr(3)p 1863 2340 192 4 v 1901 2416 a Fs(2)p Fj(c)1979 2428 y Fr(2)2083 2359 y Fs(+)2166 2206 y Fg(s)p 2249 2206 212 4 v 2259 2303 a Fs(8)p Fj(\016)2341 2279 y Fr(3)2378 2303 y Fj(c)2414 2315 y Fr(2)p 2259 2340 192 4 v 2273 2416 a Fs(27)p Fj(a)2401 2428 y Fr(1)2461 2359 y Fh(k)p Fj(u)p Fh(k)2593 2324 y Fr(3)2593 2381 y Fi(L)2639 2365 y Fe(3)2669 2381 y Fr(\(0)p Fi(;)p Fr(1\))2811 2359 y Fj(:)341 b Fs(\(4.12\))523 2600 y(Let)28 b(us)f(observ)n(e)f(that)i(on) f(the)h(attractor)701 2799 y Fj(a)745 2811 y Fr(0)801 2799 y Fs(+)894 2743 y Fj(a)938 2755 y Fr(1)975 2743 y Fj(c)1011 2755 y Fr(1)p 894 2780 155 4 v 934 2856 a Fj(c)970 2868 y Fr(2)1076 2799 y Fs(+)18 b(3)1201 2698 y Fg(q)p 1284 2698 256 4 v 101 x Fj(a)1328 2771 y Fr(2)1328 2822 y(3)1383 2799 y Fs(+)g Fj(b)1502 2771 y Fr(2)1502 2822 y(3)1539 2799 y Fh(k)p Fj(u)p Fh(k)1671 2765 y Fr(2)1730 2799 y Fh(\024)23 b Fj(a)1862 2811 y Fr(0)1917 2799 y Fs(+)2010 2743 y Fj(a)2054 2755 y Fr(1)2091 2743 y Fj(c)2127 2755 y Fr(1)p 2010 2780 155 4 v 2051 2856 a Fj(c)2087 2868 y Fr(2)2193 2799 y Fs(+)18 b(3)2318 2698 y Fg(q)p 2401 2698 256 4 v 101 x Fj(a)2445 2771 y Fr(2)2445 2822 y(3)2500 2799 y Fs(+)g Fj(b)2619 2771 y Fr(2)2619 2822 y(3)2656 2799 y Fj(\032)2699 2765 y Fr(2)2699 2820 y(0)2759 2799 y Fs(:=)23 b Fj(K)33 b(:)178 b Fs(\(4.13\))523 3018 y(But)28 b(from)f(Y)-7 b(oung's)27 b(inequalit)n(y)g(w)n(e)h(ha)n(v)n (e)1433 3256 y Fj(K)6 b(m)23 b Fh(\024)1703 3200 y Fj(a)1747 3212 y Fr(1)1784 3200 y Fj(m)1857 3170 y Fr(3)p 1703 3237 192 4 v 1742 3313 a Fs(4)p Fj(c)1820 3325 y Fr(2)1923 3256 y Fs(+)2006 3132 y Fg(r)p 2089 3132 185 4 v 2103 3200 a Fs(16)p Fj(c)2223 3212 y Fr(2)p 2099 3237 165 4 v 2099 3313 a Fs(27)p Fj(a)2227 3325 y Fr(1)2273 3256 y Fj(K)2350 3222 y Fr(3)p Fi(=)p Fr(2)2454 3256 y Fj(:)523 3481 y Fs(Therefore)j(b)n(y)i(inserting)f(the)h(ab)r(o)n(v)n(e)e (inequalit)n(y)h(in)h(\(4)p Fj(:)p Fs(12\),)f(w)n(e)g(get)609 3650 y Fi(m)579 3674 y Fg(X)581 3851 y Fi(j)s Fr(=1)713 3753 y Fj(R)q(e)p Fs(\()p Fj(F)913 3719 y Ff(0)935 3753 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1190 3765 y Fi(j)1226 3753 y Fj(;)14 b(\036)1312 3765 y Fi(j)1348 3753 y Fs(\))23 b Fh(\024)g(\000)1582 3697 y Fj(a)1626 3709 y Fr(1)p 1565 3734 115 4 v 1565 3810 a Fs(4)p Fj(c)1643 3822 y Fr(2)1690 3753 y Fj(m)1763 3719 y Fr(3)1819 3753 y Fs(+)1902 3629 y Fg(r)p 1985 3629 185 4 v 1999 3697 a Fs(16)p Fj(c)2119 3709 y Fr(2)p 1995 3734 165 4 v 1995 3810 a Fs(27)p Fj(a)2123 3822 y Fr(1)2169 3753 y Fj(K)2246 3719 y Fr(3)p Fi(=)p Fr(2)2368 3753 y Fs(+)2451 3600 y Fg(s)p 2534 3600 212 4 v 2544 3697 a Fs(8)p Fj(\016)2626 3673 y Fr(3)2663 3697 y Fj(c)2699 3709 y Fr(2)p 2544 3734 192 4 v 2558 3810 a Fs(27)p Fj(a)2686 3822 y Fr(1)2746 3753 y Fh(k)p Fj(u)p Fh(k)2878 3719 y Fr(3)2878 3776 y Fi(L)2924 3759 y Fe(3)2954 3776 y Fr(\(0)p Fi(;)p Fr(1\))3096 3753 y Fj(:)56 b Fs(\(4.14\))523 4020 y(Because)21 b(of)i(the)g(nonlo)r (cal)e(equation)h(p)r(ossesses)f(the)i(absorbing)d(balls)i(in)h(v)-5 b(arious)21 b(norms,)523 4119 y(for)27 b(some)g Fj(s)h Fs(large)e(enough)h(equation)g(\(4)p Fj(:)p Fs(14\))g(can)g(b)r(e)h (estimated)g(as)706 4271 y Fi(m)675 4296 y Fg(X)678 4473 y Fi(j)s Fr(=1)809 4375 y Fj(R)q(e)p Fs(\()p Fj(F)1009 4341 y Ff(0)1032 4375 y Fs(\()p Fj(u)p Fs(\()p Fj(t)p Fs(\)\))p Fj(\036)1287 4387 y Fi(j)1323 4375 y Fj(;)14 b(\036)1409 4387 y Fi(j)1445 4375 y Fs(\))83 b Fh(\024)1707 4251 y Fg(r)p 1790 4251 185 4 v 1804 4319 a Fs(16)p Fj(c)1924 4331 y Fr(2)p 1800 4356 165 4 v 1800 4432 a Fs(27)p Fj(a)1928 4444 y Fr(1)1988 4258 y Fg(\022)2049 4375 y Fj(a)2093 4387 y Fr(0)2149 4375 y Fs(+)2242 4319 y Fj(a)2286 4331 y Fr(1)2323 4319 y Fj(c)2359 4331 y Fr(1)p 2242 4356 155 4 v 2283 4432 a Fj(c)2319 4444 y Fr(2)2425 4375 y Fs(+)18 b(3)2550 4274 y Fg(q)p 2632 4274 256 4 v 2632 4375 a Fj(a)2676 4346 y Fr(2)2676 4397 y(3)2732 4375 y Fs(+)g Fj(b)2851 4346 y Fr(2)2851 4397 y(3)2888 4375 y Fj(\032)2931 4341 y Fr(2)2931 4395 y(0)2968 4258 y Fg(\023)3029 4275 y Fr(3)p Fi(=)p Fr(2)3152 4375 y Fs(+)1707 4532 y Fg(s)p 1790 4532 212 4 v 1800 4629 a Fs(8)p Fj(\016)1882 4605 y Fr(3)1919 4629 y Fj(c)1955 4641 y Fr(2)p 1800 4666 192 4 v 1814 4742 a Fs(27)p Fj(a)1942 4754 y Fr(1)2002 4685 y Fj(\032)2045 4651 y Fr(3)2045 4706 y Ff(1)2134 4685 y Fh(\000)2244 4629 y Fj(a)2288 4641 y Fr(1)p 2227 4666 115 4 v 2227 4742 a Fs(4)p Fj(c)2305 4754 y Fr(2)2352 4685 y Fj(m)2425 4651 y Fr(3)2462 4685 y Fj(:)690 b Fs(\(4.15\))1914 5147 y(12)p eop %%Page: 13 13 13 12 bop 523 614 a Fs(where)27 b Fj(\032)806 626 y Fr(0)871 614 y Fs(and)g Fj(\032)1075 626 y Ff(1)1173 614 y Fs(are)g(de\014ned)h (in)g(\(3)p Fj(:)p Fs(2\))f(and)h(\(3)p Fj(:)p Fs(13\)-\(3)p Fj(:)p Fs(14\))d(resp)r(ectiv)n(ely)-7 b(.)36 b(Let)789 836 y Fj(q)826 848 y Fi(m)912 836 y Fs(=)22 b(lim)14 b(sup)1048 902 y Fi(t)p Ff(!1)1390 836 y Fs(sup)1277 907 y Fd(\030)1305 920 y(j)1336 907 y Fc(2)p Fd(L)1415 893 y Fe(2)1415 924 y Fd(per)1507 907 y Fe(\(0)p Fd(;)p Fe(1\))1356 973 y Fc(k)p Fd(\030)1415 986 y(j)1445 973 y Fc(k\024)p Fe(1)1307 1033 y Fd(j)r Fe(=1)p Fd(;)p Fe(2)p Fd(;:::)q(;m)1662 780 y Fs(1)p 1662 817 42 4 v 1668 893 a Fj(t)1727 723 y Fg(Z)1810 744 y Fi(t)1773 912 y Fr(0)1853 836 y Fj(R)q(e)p Fh(f)p Fs(T)-7 b(rac)n(\()p Fj(F)2258 802 y Ff(0)2281 836 y Fs(\()p Fj(u)p Fs(\()p Fj(s)p Fs(\)\))p Fj(P)2549 848 y Fi(m)2613 836 y Fs(\()p Fj(s)p Fs(\)\))p Fh(g)p Fj(ds;)280 b Fs(\(4.16\))523 1180 y(it)28 b(follo)n(ws)f(that) 1627 1279 y Fj(q)1664 1291 y Fi(m)1750 1279 y Fh(\024)22 b(\000)p Fj(\024)1950 1291 y Fr(1)1987 1279 y Fj(m)2060 1245 y Fr(3)2116 1279 y Fs(+)c Fj(\024)2247 1291 y Fr(2)3175 1279 y Fs(\(4.17\))523 1418 y(where)1790 1517 y Fj(\024)1838 1529 y Fr(1)1898 1517 y Fs(=)2012 1461 y Fj(a)2056 1473 y Fr(1)p 1996 1498 115 4 v 1996 1574 a Fs(4)p Fj(c)2074 1586 y Fr(2)3175 1517 y Fs(\(4.18\))523 1692 y(and)808 1924 y Fj(\024)856 1936 y Fr(2)916 1924 y Fs(=)1003 1800 y Fg(r)p 1086 1800 185 4 v 1100 1868 a Fs(16)p Fj(c)1220 1880 y Fr(2)p 1096 1905 165 4 v 1096 1981 a Fs(27)p Fj(a)1224 1993 y Fr(1)1270 1924 y Fs(\()p Fj(a)1346 1936 y Fr(0)1402 1924 y Fs(+)1495 1868 y Fj(a)1539 1880 y Fr(1)1576 1868 y Fj(c)1612 1880 y Fr(1)p 1495 1905 155 4 v 1536 1981 a Fj(c)1572 1993 y Fr(2)1678 1924 y Fs(+)g(3)1803 1823 y Fg(q)p 1886 1823 256 4 v 101 x Fj(a)1930 1896 y Fr(2)1930 1947 y(3)1985 1924 y Fs(+)g Fj(b)2104 1896 y Fr(2)2104 1947 y(3)2141 1924 y Fj(\032)2184 1890 y Fr(2)2184 1945 y(0)2221 1924 y Fs(\))2253 1890 y Fr(3)p Fi(=)p Fr(2)2376 1924 y Fs(+)2459 1771 y Fg(s)p 2542 1771 212 4 v 2552 1868 a Fs(8)p Fj(\016)2634 1844 y Fr(3)2671 1868 y Fj(c)2707 1880 y Fr(2)p 2552 1905 192 4 v 2566 1981 a Fs(27)p Fj(a)2694 1993 y Fr(1)2754 1924 y Fj(\032)2797 1890 y Fr(3)2797 1945 y Ff(1)2868 1924 y Fj(:)284 b Fs(\(4.19\))523 2152 y(F)-7 b(ollo)n(wing)22 b(the)i(w)n(ork)e(of)i(Constan)n(tin,)g(F)-7 b(oias)22 b(and)i(T)-7 b(emam)23 b([1985])f(\(see)h(also)g(Constan)n (tin)523 2252 y(and)28 b(F)-7 b(oias)27 b([1988],)f(T)-7 b(emam)28 b([1988])e(and)i(references)f(therein\),)h(the)h(global)e (attractor)f Fh(A)523 2352 y Fs(has)j(Hausdor\013)g(dimension)g Fj(d)1491 2364 y Fi(H)1555 2352 y Fs(\()p Fh(A)p Fs(\))h(less)f(than)h (or)e(equal)h(to)h Fj(m)f Fs(and)h(fractal)e(dimension)523 2451 y Fj(d)566 2463 y Fi(F)621 2451 y Fs(\()p Fh(A)p Fs(\))h(less)e(than)h(or)f(equal)g(to)g(2)p Fj(m)g Fs(where)g Fj(m)h Fs(is)f(de\014ned)h(as)1502 2689 y Fj(m)18 b Fh(\000)g Fs(1)23 b Fh(\024)1829 2572 y Fg(\022)1900 2632 y Fs(2)p Fj(\024)1990 2644 y Fr(2)p 1900 2670 127 4 v 1920 2746 a Fj(\024)1968 2758 y Fr(1)2036 2572 y Fg(\023)2097 2589 y Fr(1)p Fi(=)p Fr(3)2225 2689 y Fh(\024)f Fj(m:)767 b Fs(\(4.20\))523 2903 y(Therefore,)26 b(using)i(\(4)p Fj(:)p Fs(18\))f(and)g(\(4)p Fj(:)p Fs(19\))g(w)n(e)g(\014nd)h(that) 1081 3128 y Fj(d)1124 3140 y Fi(H)1187 3128 y Fs(\()p Fh(A)p Fs(\))c Fh(\024)e Fj(d)1471 3140 y Fi(F)1527 3128 y Fs(\()p Fh(A)p Fs(\))i(=)e(4)1810 3003 y Fg(r)p 1893 3003 143 4 v 1907 3071 a Fs(2)p Fj(c)1985 3083 y Fr(2)p 1903 3108 123 4 v 1903 3184 a Fs(3)p Fj(a)1989 3196 y Fr(1)2049 3035 y Fg(\020)2099 3128 y Fj(K)2176 3093 y Fr(3)p Fi(=)p Fr(2)2298 3128 y Fs(+)c Fj(\016)2421 3093 y Fr(3)p Fi(=)p Fr(2)2526 3128 y Fj(\032)2569 3093 y Fr(3)2569 3148 y Ff(1)2639 3035 y Fg(\021)2688 3053 y Fr(1)p Fi(=)p Fr(3)2807 3128 y Fj(:)345 b Fs(\(4.21\))523 3335 y(where)36 b Fj(K)q(;)14 b(\016)o(;)g(\032)997 3347 y Ff(1)1102 3335 y Fs(are)36 b(de\014ned)g(in)h(\(4)p Fj(:)p Fs(13\),)g(\(3)p Fj(:)p Fs(3\))f(and)g(\(3)p Fj(:)p Fs(13\)-\(3)p Fj(:)p Fs(14\))e(resp)r(ectiv)n(ely)-7 b(.)62 b(As)523 3434 y(w)n(e)34 b(can)g(see,)h(the)g(upp)r(er)f(b)r (ound)h(\(4)p Fj(:)p Fs(21\))f(for)f(the)i(dimension)f(of)g(the)h (global)e(attractor)523 3534 y(of)i(the)g(nonlo)r(cal)f(equation)h(is)f (v)n(ery)g(complicated.)58 b(Ho)n(w)n(ev)n(er)33 b(w)n(e)i(can)f(see)h (the)g(upp)r(er)523 3633 y(b)r(ound)30 b(c)n(hanges)f(with)h(resp)r (ect)g(to)g(the)g(co)r(e\016cien)n(ts)g(of)g(the)g(nonlo)r(cal)f(term)h Fj(a)3060 3645 y Fr(3)3128 3633 y Fs(and)f Fj(b)3327 3645 y Fr(3)3364 3633 y Fs(.)523 3733 y(Moreo)n(v)n(er)k(when)i(the)h (nonlo)r(cal)e(term)i(is)f(absen)n(t,)i(w)n(e)e(can)g(compare)f(with)h (the)h(upp)r(er)523 3833 y(b)r(ound)i(of)g(the)h(global)d(attractor)h (of)g(the)i(usual)e(cubic)h(Ginzburg-Landau)f(equation.)523 3932 y(This)28 b(is)f(in)n(terpreted)g(as)g(follo)n(ws:)625 4084 y(1.)41 b(In)27 b(the)h(case)f Fh(j)p Fj(b)1212 4096 y Fr(2)1249 4084 y Fh(j)c(\024)1383 4015 y(p)p 1452 4015 42 4 v 69 x Fs(3)o Fh(j)p Fj(a)1560 4096 y Fr(2)1598 4084 y Fh(j)k Fs(\(i.e.)38 b Fj(\016)26 b Fs(=)c(0\),)1303 4346 y Fj(d)1346 4358 y Fi(H)1409 4346 y Fs(\()p Fh(A)p Fs(\))i Fh(\024)e Fj(c)1686 4312 y Ff(0)1686 4367 y Fr(1)1737 4204 y Fg( )1803 4346 y Fs(1)c(+)1956 4290 y Fj(a)2000 4302 y Fr(0)p 1956 4327 82 4 v 1956 4403 a Fj(a)2000 4415 y Fr(1)2066 4346 y Fs(+)2159 4290 y Fj(a)2203 4302 y Fr(0)p 2159 4327 V 2159 4403 a Fj(a)2203 4415 y Fr(1)2260 4217 y Fg(p)p 2343 4217 256 4 v 72 x Fj(a)2387 4261 y Fr(2)2387 4312 y(3)2442 4289 y Fs(+)g Fj(b)2561 4261 y Fr(2)2561 4312 y(3)p 2260 4327 339 4 v 2274 4403 a Fh(j)p Fj(a)2341 4415 y Fr(2)2397 4403 y Fs(+)g Fj(a)2524 4415 y Fr(3)2561 4403 y Fh(j)2608 4204 y Fg(!)2674 4221 y Fr(1)p Fi(=)p Fr(2)2792 4346 y Fj(:)360 b Fs(\(4.22\))731 4581 y(F)-7 b(or)44 b(the)h(usual)g(cubic)g(Ginzburg-Landau)e (equation,)49 b(Ghidaglia)44 b(and)h(H)n(\023)-39 b(eron)731 4680 y([1987])25 b(found)1588 4853 y Fj(d)1631 4865 y Fi(H)1694 4853 y Fs(\()p Fh(A)p Fs(\))f Fh(\024)f Fj(c)1972 4818 y Ff(0)1972 4873 y Fr(1)2023 4736 y Fg(\022)2084 4853 y Fs(1)18 b(+)2237 4796 y Fj(a)2281 4808 y Fr(0)p 2237 4834 82 4 v 2237 4910 a Fj(a)2281 4922 y Fr(1)2328 4736 y Fg(\023)2389 4753 y Fr(1)p Fi(=)p Fr(2)2507 4853 y Fj(:)645 b Fs(\(4.23\))1914 5147 y(13)p eop %%Page: 14 14 14 13 bop 731 614 a Fs(Therefore)25 b(b)n(y)h(comparing)f(with)j(the)f (usual)f(cubic)h(Ginzburg-Landau)e(equation,)731 714 y(w)n(e)33 b(\014nd)h(that)f(the)h(di\013erence)f(b)r(et)n(w)n(een)h (the)g(square)e(of)h(the)h(the)g(upp)r(er)f(b)r(ounds)731 814 y(of)27 b(the)g(dimensions)g(of)g(the)h(global)e(attractor)f Fh(A)j Fs(for)e(the)i(nonlo)r(cal)e(equation)h(and)731 913 y(the)h(cubic)f(Ginzburg-Landau)f(is)i(equal)f(to)1801 1100 y Fj(c)1837 1069 y Ff(0)1837 1120 y Fr(1)1874 1100 y Fj(a)1918 1112 y Fr(0)1955 1027 y Fg(p)p 2038 1027 256 4 v 73 x Fj(a)2082 1071 y Fr(2)2082 1122 y(3)2138 1100 y Fs(+)18 b Fj(b)2257 1071 y Fr(2)2257 1122 y(3)p 1801 1137 493 4 v 1852 1213 a Fj(a)1896 1225 y Fr(1)1933 1213 y Fh(j)p Fj(a)2000 1225 y Fr(2)2056 1213 y Fs(+)g Fj(a)2183 1225 y Fr(3)2220 1213 y Fh(j)2304 1156 y Fj(:)848 b Fs(\(4.24\))625 1434 y(2.)41 b(F)-7 b(or)28 b(the)h(case)f Fh(j)p Fj(b)1261 1446 y Fr(2)1298 1434 y Fh(j)d Fj(>)1435 1365 y Fh(p)p 1505 1365 42 4 v 1505 1434 a Fs(3)o Fh(j)p Fj(a)1613 1446 y Fr(2)1650 1434 y Fh(j)k Fs(\(i.e.)41 b Fj(\016)28 b(>)d Fs(0\),)k(w)n(e)f(can)h(see)f(from)g(\(4)p Fj(:)p Fs(20\))h(that)g(along)731 1533 y(with)37 b(the)g(increase)e(of) h(\(4)p Fj(:)p Fs(24\))g(as)g(in)h(case)e(1)i(ab)r(o)n(v)n(e,)g(the)g (upp)r(er)g(b)r(ound)g(of)f(the)731 1633 y(dimension)25 b(of)h(the)h(global)e(attractor)f Fh(A)i Fs(for)g(the)g(nonlo)r(cal)f (equation)h(is)g(increased)731 1732 y(further)h(due)h(to)f(the)h(c)n (hange)f(of)g Fj(\032)1823 1744 y Ff(1)1894 1732 y Fs(.)37 b(That)27 b(is)965 2010 y Fj(d)1008 2022 y Fi(H)1071 2010 y Fs(\()p Fh(A)p Fs(\))84 b Fh(\024)f Fj(c)1469 1976 y Ff(00)1469 2031 y Fr(1)1525 1840 y Fg(8)1525 1915 y(<)1525 2064 y(:)1599 1868 y( )1665 2010 y Fs(1)18 b(+)1818 1954 y Fj(a)1862 1966 y Fr(0)p 1818 1991 82 4 v 1818 2067 a Fj(a)1862 2079 y Fr(1)1927 2010 y Fs(+)2020 1954 y Fj(a)2064 1966 y Fr(0)2101 1881 y Fg(p)p 2184 1881 256 4 v 73 x Fj(a)2228 1925 y Fr(2)2228 1976 y(3)2284 1954 y Fs(+)g Fj(b)2403 1925 y Fr(2)2403 1976 y(3)p 2020 1991 420 4 v 2034 2067 a Fj(a)2078 2079 y Fr(1)2116 2067 y Fh(j)p Fj(a)2183 2079 y Fr(2)2238 2067 y Fs(+)g Fj(a)2365 2079 y Fr(3)2402 2067 y Fh(j)2450 1868 y Fg(!)2515 1885 y Fr(1)p Fi(=)p Fr(2)2634 2010 y Fs(+)1433 2175 y Fg(\022)1639 2236 y Fj(a)1683 2248 y Fr(0)1720 2236 y Fj(\016)p 1504 2273 391 4 v 1504 2349 a(a)1548 2361 y Fr(1)1585 2349 y Fh(j)p Fj(a)1652 2361 y Fr(2)1708 2349 y Fs(+)g Fj(a)1835 2361 y Fr(3)1872 2349 y Fh(j)1919 2175 y Fg(\022)1980 2292 y Fs(1)g(+)2247 2236 y(2)p Fj(\016)p 2133 2273 310 4 v 2133 2349 a Fh(j)p Fj(a)2200 2361 y Fr(2)2256 2349 y Fs(+)g Fj(a)2383 2361 y Fr(3)2420 2349 y Fh(j)2467 2292 y Fs([1)g(+)g(\(1+)1493 2519 y Fj(a)1537 2531 y Fr(1)1574 2519 y Fh(j)p Fj(a)1641 2531 y Fr(2)1696 2519 y Fs(+)h Fj(a)1824 2531 y Fr(3)1861 2519 y Fh(j)p Fs(\()p Fh(j)p Fj(a)1983 2531 y Fr(2)2039 2519 y Fs(+)f Fj(a)2166 2531 y Fr(3)2203 2519 y Fh(j)g Fs(+)g Fj(\016)k Fs(+)c Fj(a)2513 2531 y Fr(3)2550 2519 y Fs(\))p 1493 2556 1090 4 v 1958 2632 a Fj(\016)1998 2608 y Fr(2)2036 2632 y Fj(a)2080 2644 y Fr(0)2592 2458 y Fg(\023)2653 2475 y Fr(1)p Fi(=)p Fr(2)2758 2433 y Fg(#!!)2938 2450 y Fr(1)p Fi(=)p Fr(2)3042 2404 y Fg(9)3042 2479 y(=)3042 2629 y(;)3130 2575 y Fj(:)k Fs(\(4.25\))648 2886 y(W)-7 b(e)28 b(no)n(w)g(establish)g(the)g(lo)n(w)n(er)f(b)r(ound)h(of)g(the)h (dimension)f(of)g(the)h(global)e(attractor.)523 2985 y(F)-7 b(ollo)n(wing)37 b(Ghidaglia)h(and)h(H)n(\023)-39 b(eron)37 b([1987],)i(w)n(e)g(linearize)e(the)i(equation)f(around)g (the)523 3085 y(Stok)n(es)27 b(solution)1311 3294 y Fj(u)1359 3306 y Fi(s)1394 3294 y Fs(\()p Fj(t)p Fs(\))c(=)1599 3183 y Fg(r)p 1682 3183 330 4 v 1806 3238 a Fj(a)1850 3250 y Fr(0)p 1692 3275 310 4 v 1692 3351 a Fh(j)p Fj(a)1759 3363 y Fr(2)1815 3351 y Fs(+)18 b Fj(a)1942 3363 y Fr(3)1979 3351 y Fh(j)2012 3294 y Fj(e)2051 3253 y Fi(ia)2110 3261 y Fe(0)2152 3221 y(\()p Fd(b)2200 3233 y Fe(2)2234 3221 y(+)p Fd(b)2303 3233 y Fe(3)2335 3221 y(\))p 2152 3240 206 4 v 2169 3273 a Fd(a)2201 3285 y Fe(2)2234 3273 y(+)p Fd(a)2309 3285 y Fe(3)2367 3253 y Fi(t)2397 3294 y Fj(;)755 b Fs(\(4.26\))523 3528 y(and)30 b(c)n(hec)n(k)g(the)g(n)n(um)n(b)r(er)g (of)h(unstable)f(eigen)n(v)-5 b(alues)29 b(for)h(the)h(corresp)r (onding)d(linearized)523 3627 y(di\013eren)n(tial)34 b(op)r(erator.)55 b(This)34 b(n)n(um)n(b)r(er)g(is)h(the)f(dimension)g (of)h(the)f(in)n(v)-5 b(arian)n(t)33 b(unstable)523 3727 y(manifold)d Fj(E)930 3739 y Fr(+)986 3727 y Fs(\()p Fj(u)1066 3739 y Fi(s)1101 3727 y Fs(\()p Fj(t)p Fs(\)\).)45 b(According)29 b(to)h(Theorem)f(VI)r(I)i(3.2)e(of)h(T)-7 b(emam)30 b([1988])e(\(see)i(also)523 3827 y(Babin)i(and)f(Vishik)h ([1992])e(and)i(references)f(therein\))h Fj(E)2374 3839 y Fr(+)2429 3827 y Fs(\()p Fj(u)2509 3839 y Fi(s)2544 3827 y Fs(\()p Fj(t)p Fs(\)\))h(is)f(con)n(tained)f(in)h(the)523 3926 y(global)d(attractor)f Fh(A)p Fs(,)i(and)g(hence)g(the)g (dimension)f(of)h Fj(E)2336 3938 y Fr(+)2391 3926 y Fs(\()p Fj(u)2471 3938 y Fi(s)2507 3926 y Fs(\()p Fj(t)p Fs(\)\))g(giv)n(es)f (a)g(lo)n(w)n(er)f(b)r(ound)523 4026 y(of)g(the)g(dimension)f(of)h Fh(A)p Fs(.)648 4125 y(W)-7 b(e)28 b(mak)n(e)e(the)i(follo)n(wing)f(c)n (hange)f(of)i(v)-5 b(ariables)1601 4308 y Fj(u)22 b Fs(=)h Fj(u)1807 4320 y Fi(s)1842 4308 y Fs(\(1)18 b(+)g Fj(v)s Fs(\))28 b Fj(;)523 4491 y Fs(and)j(write)f(the)h(equation)f(\(1.1\))h (in)g(terms)f(of)h Fj(v)s Fs(.)46 b(W)-7 b(e)31 b(then)h(linearize)d (this)i(equation)f(at)523 4590 y Fj(v)c Fs(=)d(0)k(in)h(the)g (direction)f(of)h Fj(w)i Fs(to)e(obtain:)1630 4773 y Fj(w)1689 4785 y Fi(t)1737 4773 y Fs(+)18 b Fj(Lw)26 b Fs(=)c(0)p Fj(;)1914 5147 y Fs(14)p eop %%Page: 15 15 15 14 bop 523 614 a Fs(where)27 b(the)h(linear)f(op)r(erator)f Fj(L)h Fs(is)g(giv)n(en)g(b)n(y:)644 836 y Fj(Lw)e Fs(=)e(\()p Fj(a)949 848 y Fr(1)1005 836 y Fs(+)18 b Fj(ib)1153 848 y Fr(1)1189 836 y Fs(\)[)p Fj(Aw)k Fs(+)1480 780 y Fj(a)c Fs(+)g Fj(ib)p 1480 817 210 4 v 1564 893 a Fs(2)1699 836 y(\()p Fj(w)k Fs(+)35 b(\026)-59 b Fj(w)s Fs(\)])18 b(+)2123 780 y Fj(a)2167 792 y Fr(0)2204 780 y Fs(\()p Fj(a)2280 792 y Fr(3)2336 780 y Fs(+)g Fj(ib)2484 792 y Fr(3)2521 780 y Fs(\))p 2123 817 430 4 v 2183 893 a Fh(j)p Fj(a)2250 905 y Fr(2)2306 893 y Fs(+)g Fj(a)2433 905 y Fr(3)2470 893 y Fh(j)2577 723 y Fg(Z)2660 743 y Fr(1)2623 911 y(0)2697 836 y Fs(\()p Fj(w)k Fs(+)35 b(\026)-59 b Fj(w)r Fs(\))p Fj(dx;)523 1046 y Fs(with)28 b(the)g(constan)n(ts)1400 1236 y Fj(a)23 b Fs(=)1658 1179 y(2)p Fj(a)1744 1191 y Fr(0)p 1564 1217 310 4 v 1564 1293 a Fh(j)p Fj(a)1631 1305 y Fr(2)1687 1293 y Fs(+)18 b Fj(a)1814 1305 y Fr(3)1851 1293 y Fh(j)1894 1179 y Fj(a)1938 1191 y Fr(1)1975 1179 y Fj(a)2019 1191 y Fr(2)2075 1179 y Fs(+)g Fj(b)2194 1191 y Fr(1)2231 1179 y Fj(b)2267 1191 y Fr(2)p 1894 1217 410 4 v 1971 1293 a Fj(a)2015 1264 y Fr(2)2015 1315 y(1)2071 1293 y Fs(+)g Fj(b)2190 1264 y Fr(2)2190 1315 y(1)2313 1236 y Fj(;)1394 1462 y(b)23 b Fs(=)1644 1406 y(2)p Fj(a)1730 1418 y Fr(0)p 1550 1443 310 4 v 1550 1519 a Fh(j)p Fj(a)1617 1531 y Fr(2)1673 1519 y Fs(+)18 b Fj(a)1800 1531 y Fr(3)1837 1519 y Fh(j)1880 1406 y Fj(a)1924 1418 y Fr(1)1961 1406 y Fj(b)1997 1418 y Fr(2)2053 1406 y Fh(\000)g Fj(b)2172 1418 y Fr(1)2208 1406 y Fj(a)2252 1418 y Fr(2)p 1880 1443 410 4 v 1957 1519 a Fj(a)2001 1490 y Fr(2)2001 1541 y(1)2057 1519 y Fs(+)g Fj(b)2176 1490 y Fr(2)2176 1541 y(1)2327 1462 y Fj(:)523 1674 y Fs(W)-7 b(e)20 b(need)g(to)f(estimate)g(the)h(n)n(um)n(b)r(er)g(of)f (unstable)h(eigenfunctions)f(of)g(the)h(linear)f(op)r(erator)523 1773 y Fj(L)p Fs(,)35 b(i.e.,)h(those)e(whic)n(h)g(corresp)r(ond)e(to)i (eigen)n(v)-5 b(alues)33 b(with)i(p)r(ositiv)n(e)e(real)g(part.)56 b(Let)34 b(us)523 1882 y(denote)41 b(b)n(y)g Fj(w)48 b Fs(=)d Fj(R)28 b Fs(+)f Fj(iI)7 b Fs(,)44 b(and)c(b)n(y)h Fj(w)1834 1894 y Fr(0)1917 1882 y Fs(=)k Fj(R)2090 1894 y Fr(0)2155 1882 y Fs(+)27 b Fj(iI)2312 1894 y Fr(0)2394 1882 y Fs(=)2504 1815 y Fg(R)2560 1835 y Fr(1)2543 1911 y(0)2611 1882 y Fj(w)r Fs(\()p Fj(x)p Fs(\))p Fj(dx)p Fs(.)79 b(The)41 b Fj(Lw)48 b Fs(=)523 1981 y(\()p Fj(a)599 1993 y Fr(1)655 1981 y Fs(+)18 b Fj(ib)803 1993 y Fr(1)839 1981 y Fs(\))p Fj(T)12 b(w)r Fs(,)28 b(where)1092 2196 y Fj(T)12 b(w)25 b Fs(=)d Fj(Aw)g Fs(+)c(\()p Fj(a)g Fs(+)g Fj(ib)p Fs(\))p Fj(R)h Fs(+)2113 2140 y Fj(a)2157 2152 y Fr(0)2194 2140 y Fs(\()p Fj(a)2270 2152 y Fr(3)2326 2140 y Fs(+)f Fj(ib)2474 2152 y Fr(3)2511 2140 y Fs(\))p 1999 2177 659 4 v 1999 2253 a Fh(j)p Fj(a)2066 2265 y Fr(2)2122 2253 y Fs(+)g Fj(a)2249 2265 y Fr(3)2286 2253 y Fh(j)p Fs(\()p Fj(a)2385 2265 y Fr(1)2441 2253 y Fs(+)g Fj(ib)2589 2265 y Fr(1)2625 2253 y Fs(\))2667 2196 y Fj(R)2730 2208 y Fr(0)2795 2196 y Fj(:)523 2407 y Fs(This)24 b(should)h(b)r(e)f(understo)r(o)r(d)g(as)g(a)g(real)g(op)r(erator)e (de\014ned)j(on)f(the)h(space)e(of)i(v)n(ector)e(v)-5 b(al-)523 2506 y(ued)24 b(functions)h(o)n(v)n(er)d(the)i(scalar)f (\014eld)h(of)g(real)f(n)n(um)n(b)r(ers.)36 b(Namely)-7 b(,)24 b(as)g(a)f(map)h Fj(T)12 b Fs(\()p Fj(R)q(;)i(I)7 b Fs(\))23 b(=)523 2631 y(\()p Fj(AR)10 b Fs(+)f Fj(aR)h Fs(+)f Fj(R)q(e)1072 2539 y Fg(n)1215 2590 y Fi(a)1251 2598 y Fe(0)1284 2590 y Fr(\()p Fi(a)1346 2598 y Fe(3)1378 2590 y Fr(+)p Fi(ib)1481 2598 y Fe(3)1514 2590 y Fr(\))p 1136 2612 484 4 v 1136 2659 a Ff(j)p Fi(a)1192 2667 y Fe(2)1224 2659 y Fr(+)p Fi(a)1311 2667 y Fe(3)1343 2659 y Ff(j)p Fr(\()p Fi(a)1425 2667 y Fe(1)1457 2659 y Fr(+)p Fi(ib)1560 2667 y Fe(1)1593 2659 y Fr(\))1629 2539 y Fg(o)1698 2631 y Fj(R)1761 2643 y Fr(0)1799 2631 y Fj(;)14 b(AI)i Fs(+)9 b Fj(bR)h Fs(+)f Fj(I)e(m)2337 2539 y Fg(n)2480 2590 y Fi(a)2516 2598 y Fe(0)2548 2590 y Fr(\()p Fi(a)2610 2598 y Fe(3)2643 2590 y Fr(+)p Fi(ib)2746 2598 y Fe(3)2779 2590 y Fr(\))p 2401 2612 V 2401 2659 a Ff(j)p Fi(a)2457 2667 y Fe(2)2489 2659 y Fr(+)p Fi(a)2576 2667 y Fe(3)2608 2659 y Ff(j)p Fr(\()p Fi(a)2690 2667 y Fe(1)2722 2659 y Fr(+)p Fi(ib)2825 2667 y Fe(1)2858 2659 y Fr(\))2894 2539 y Fg(o)2963 2631 y Fj(R)3026 2643 y Fr(0)3064 2631 y Fs(\).)35 b(Notice)523 2755 y(that)23 b(sp)r(ec\()p Fj(L)p Fs(\))h(=)e(\()p Fj(a)1161 2767 y Fr(1)1208 2755 y Fs(+)10 b Fj(ib)1348 2767 y Fr(1)1384 2755 y Fs(\)sp)r(ec\()p Fj(T)i Fs(\).)35 b(Therefore)22 b(it)i(is)f(enough)f(to)h(\014nd)h(the) f(sp)r(ectrum)g(of)523 2855 y Fj(T)12 b Fs(.)34 b(It)21 b(is)g(easy)f(to)h(see)g(that)g(the)h(space)e(of)h(constan)n(t)f(v)n (ector)g(v)-5 b(alued)21 b(functions)h(is)f(in)n(v)-5 b(arian)n(t)523 2955 y(subspace)22 b(under)g Fj(T)12 b Fs(,)23 b(and)f(that)h Fj(T)34 b Fs(has)22 b(only)g(t)n(w)n(o)f (constan)n(t)h(linearly)g(indep)r(enden)n(t)h(eigen-)523 3074 y(functions)c(corresp)r(onding)d(to)i(the)h(eigen)n(v)-5 b(alues)17 b Fh(f)p Fj(\026)23 b Fs(=)g(0)p Fj(;)14 b(\026)22 b Fs(=)h Fj(a)p Fs(+)p Fj(R)q(e)2710 2982 y Fg(n)2853 3034 y Fi(a)2889 3042 y Fe(0)2922 3034 y Fr(\()p Fi(a)2984 3042 y Fe(3)3016 3034 y Fr(+)p Fi(ib)3119 3042 y Fe(3)3152 3034 y Fr(\))p 2774 3055 V 2774 3103 a Ff(j)p Fi(a)2830 3111 y Fe(2)2862 3103 y Fr(+)p Fi(a)2949 3111 y Fe(3)2981 3103 y Ff(j)p Fr(\()p Fi(a)3063 3111 y Fe(1)3095 3103 y Fr(+)p Fi(ib)3198 3111 y Fe(1)3231 3103 y Fr(\))3267 2982 y Fg(o)3323 3074 y Fh(g)p Fs(.)523 3199 y(The)31 b(rest)f(of)h(sp)r(ectrum)g(of)f Fj(T)42 b Fs(can)30 b(b)r(e)h(found,)h(as)e(in)h(Ghidaglia)f(and)h(H)n(\023)-39 b(eron)29 b([1987],)g(to)523 3299 y(b)r(e)f Fh([)691 3268 y Ff(1)691 3320 y Fi(j)s Fr(=0)810 3299 y Fh(f)p Fj(j)891 3268 y Fr(2)928 3299 y Fj(;)14 b(j)1004 3268 y Fr(2)1060 3299 y Fs(+)k Fj(a)p Fh(g)p Fs(.)36 b(As)28 b(a)f(result,)g(w)n(e)h(ha)n(v)n(e:)703 3527 y(sp)r(ec\()p Fj(T)12 b Fs(\))23 b(=)g Fh([)1149 3493 y Ff(1)1149 3548 y Fi(j)s Fr(=0)1268 3527 y Fh(f)p Fj(j)1349 3493 y Fr(2)1386 3527 y Fj(;)14 b Fs(\()p Fj(j)1494 3493 y Fr(2)1550 3527 y Fs(+)k Fj(a)p Fs(\))p Fh(g)g([)h(f)p Fj(a)e Fs(+)h Fj(R)q(e)2146 3410 y Fg(\032)2332 3471 y Fj(a)2376 3483 y Fr(0)2413 3471 y Fs(\()p Fj(a)2489 3483 y Fr(3)2545 3471 y Fs(+)g Fj(ib)2693 3483 y Fr(3)2730 3471 y Fs(\))p 2218 3508 659 4 v 2218 3584 a Fh(j)p Fj(a)2285 3596 y Fr(2)2341 3584 y Fs(+)g Fj(a)2468 3596 y Fr(3)2505 3584 y Fh(j)p Fs(\()p Fj(a)2604 3596 y Fr(1)2660 3584 y Fs(+)g Fj(ib)2808 3596 y Fr(1)2844 3584 y Fs(\))2886 3410 y Fg(\033)2949 3527 y Fh(g)27 b Fj(:)523 3739 y Fs(W)-7 b(e)31 b(need)g(to)g(\014nd)g(the)g(n)n(um)n(b)r(er)g(of)f(eigen)n(v)-5 b(alues)30 b(of)h Fj(L)p Fs(,)g(including)g(their)g(m)n(ultiplicities,) 523 3839 y(whic)n(h)d(lie)f(in)h(the)g(left)g(half)g(complex)f(plane.) 37 b(When)28 b Fj(a)23 b(<)g Fs(0,)k(i.e.,)h(when)1580 4005 y Fj(a)1624 4017 y Fr(1)1661 4005 y Fj(a)1705 4017 y Fr(2)1760 4005 y Fs(+)19 b Fj(b)1880 4017 y Fr(1)1916 4005 y Fj(b)1952 4017 y Fr(2)2012 4005 y Fj(<)k Fs(0)p Fj(;)523 4172 y Fs(there)i(are)f(unstable)h(eigen)n(v)-5 b(alues,)24 b(and)h(as)f(in)i(Ghidaglia)e(and)h(H)n(\023)-39 b(eron)23 b([1987])g(or)h(T)-7 b(emam)523 4272 y([1988,)23 b(page)g(406])f(w)n(e)i(can)f(estimate)h(the)g(lo)n(w)n(er)f(b)r(ound)h (of)g(the)g(dimension)g(of)f(the)i(global)523 4371 y(attractor)865 4602 y Fj(dim)1010 4614 y Fi(H)1072 4602 y Fs(\()p Fh(A)p Fs(\))f Fh(\025)f Fj(dimE)1520 4614 y Fr(+)1575 4602 y Fs(\()p Fj(u)1655 4614 y Fi(s)1690 4602 y Fs(\))h Fh(\025)e(\000)p Fs(1)c(+)g(2)2083 4453 y Fg(s)p 2165 4453 650 4 v 2201 4546 a Fs(2)p Fj(a)2287 4558 y Fr(0)2324 4546 y Fh(j)p Fj(a)2391 4558 y Fr(1)2428 4546 y Fj(a)2472 4558 y Fr(2)2528 4546 y Fs(+)g Fj(b)2647 4558 y Fr(1)2683 4546 y Fj(b)2719 4558 y Fr(2)2756 4546 y Fh(j)p 2175 4583 630 4 v 2175 4659 a(j)p Fj(a)2242 4671 y Fr(2)2298 4659 y Fs(+)g Fj(a)2425 4671 y Fr(3)2462 4659 y Fh(j)p Fs(\()p Fj(a)2561 4630 y Fr(2)2561 4681 y(1)2617 4659 y Fs(+)g Fj(b)2736 4630 y Fr(2)2736 4681 y(1)2773 4659 y Fs(\))2815 4602 y Fh(\000)1199 4853 y Fs(min)p Fh(f)p Fs(0)p Fj(;)c Fs(sgn)n(\()p Fj(a)1653 4865 y Fr(1)1691 4853 y Fj(a)k Fs(+)g Fj(R)q(e)1953 4736 y Fg(\032)2098 4796 y Fj(a)2142 4808 y Fr(1)2179 4796 y Fj(a)2223 4808 y Fr(0)2260 4796 y Fs(\()p Fj(a)2336 4808 y Fr(3)2392 4796 y Fs(+)g Fj(ib)2540 4808 y Fr(3)2577 4796 y Fs(\))p 2024 4834 659 4 v 2024 4910 a Fh(j)p Fj(a)2091 4922 y Fr(2)2147 4910 y Fs(+)g Fj(a)2274 4922 y Fr(3)2311 4910 y Fh(j)p Fs(\()p Fj(a)2410 4922 y Fr(1)2466 4910 y Fs(+)g Fj(ib)2614 4922 y Fr(1)2651 4910 y Fs(\))2693 4736 y Fg(\033)2755 4853 y Fs(\))p Fh(g)28 b Fj(:)295 b Fs(\(4.27\))1914 5147 y(15)p eop %%Page: 16 16 16 15 bop 523 614 a Fs(This)34 b(is)g(a)g(lo)n(w)n(er)f(b)r(ound)i(of)f (the)h(dimension)f(of)g(the)h(global)e(attractor)f Fh(A)p Fs(.)58 b(Note)34 b(that)523 714 y(w)n(e)d(ha)n(v)n(e)e(assumed)h(that) i Fj(a)1407 726 y Fr(2)1464 714 y Fs(+)21 b Fj(a)1594 726 y Fr(3)1659 714 y Fj(<)28 b Fs(0)j(and)f Fj(a)2033 726 y Fr(2)2099 714 y Fj(<)e Fs(0.)46 b(Under)31 b(these)g(conditions,) g(when)523 814 y(compared)h(with)h(the)g(corresp)r(onding)e(cubic)i (Ginzburg-Landau)e(equation,)j(the)f(lo)n(w)n(er)523 913 y(b)r(ound)23 b(of)f(the)g(dimension)g(of)g Fh(A)h Fs(smaller)e(when)h Fj(a)2101 925 y Fr(3)2161 913 y Fj(<)h Fs(0,)g(and)f(larger)e(when)i Fj(a)2978 925 y Fr(3)3038 913 y Fj(>)h Fs(0.)34 b(This)523 1013 y(means)29 b(that)h(the)g(nonlo)r (cal)e(term)i(has)f(global)f(stabilizing)h(and)h(destabilizing)f (e\013ect)h(on)523 1112 y(the)36 b(global)f(dynamics)h(dep)r(ending)g (on)g(the)g(sign)g(of)g(the)g(real)f(part)g(of)h(its)g(co)r(e\016cien)n (t.)523 1212 y(Th)n(us)27 b(w)n(e)h(ha)n(v)n(e)e(pro)n(v)n(ed)g(the)i (follo)n(w)f(theorem.)523 1395 y Fo(Theorem)j(2)42 b Fn(Assume)30 b(that)g Fj(a)1543 1407 y Fr(1)1605 1395 y Fj(>)25 b Fs(0)p Fj(;)14 b(a)1818 1407 y Fr(2)1879 1395 y Fh(\024)25 b Fs(0)30 b Fn(and)h Fj(a)2247 1407 y Fr(2)2304 1395 y Fs(+)18 b Fj(a)2431 1407 y Fr(3)2493 1395 y Fj(<)25 b Fs(0)p Fn(.)41 b(Then)32 b(ther)l(e)f(exists)f(a)523 1494 y(\014nite)f(dimensional)i(glob)l(al)g(attr)l(actor)f(in)f Fj(L)1930 1464 y Fr(2)1930 1515 y Fi(per)2032 1494 y Fs(\(0)p Fj(;)14 b Fs(1\))29 b Fn(for)h(the)g(dynamic)l(al)h(system)e (\(1.1\)-)523 1594 y(\(1.2\)-\(1.3\).)38 b(The)23 b(upp)l(er)g(and)f (lower)h(b)l(ounds)f(for)h(the)f(dimension)h(of)g(the)f(glob)l(al)h (attr)l(actor)523 1694 y(ar)l(e)30 b(given)h(in)e(\(4.21\))j(and)e (\(4.27\),)i(r)l(esp)l(e)l(ctively.)523 1968 y Fk(5)135 b(Inertial)46 b(manifold)g(and)f(its)g(dimension)h(estimate)523 2150 y Fs(In)21 b(this)g(section)g(w)n(e)f(will)h(\014rst)g(sho)n(w)f (the)h(existence)f(of)h(inertial)f(manifold)h(of)g(the)g(nonlo)r(cal) 523 2250 y(amplitude)h(equation)f(\(1)p Fj(:)p Fs(1\),)h(then)g(w)n(e)f (will)h(giv)n(e)e(an)h(explicit)h(estimate)g(to)f(its)g(dimension.)523 2349 y(W)-7 b(e)42 b(will)f(also)g(compare)f(this)i(dimension)f (estimate)g(with)h(that)g(of)f(the)h(usual)f(cubic)523 2449 y(Ginzburg-Landau)20 b(equation;)i(in)g(order)d(to)i(\014nd)h (evidence)f(of)g(the)g(e\013ect)h(of)f(the)g(nonlo)r(cal)523 2549 y(term.)648 2648 y(The)30 b(existence)g(and)g(dimension)g (estimate)h(for)e(an)h(inertial)g(manifold)h(of)f(the)h(usual)523 2748 y(cubic)40 b(Ginzburg-Landau)f(equation)g(is)h(studied)h(in)f (Constan)n(tin)f(et)i(al.)73 b([1989])39 b(and)523 2847 y(Do)r(ering)21 b(et)g(al.)35 b([1988].)e(W)-7 b(e)21 b(are)f(going)g(to)h(pro)n(v)n(e)f(the)i(existence)e(of)i(an)f (inertial)f(manifold)523 2947 y(for)29 b(the)h(nonlo)r(cal)f(amplitude) h(equation)e(\(1.1\))i(using)f(the)h(sp)r(ectral)f(barrier)e(metho)r(d) j(of)523 3047 y(Constan)n(tin)d(et)h(al.)37 b([1989].)e(This)27 b(can)h(also)e(b)r(e)i(done)f(follo)n(wing)g(the)h(argumen)n(ts)e(in)i (Sell)523 3146 y(and)20 b(Y)-7 b(ou)19 b([1992])f(or)h(Debussc)n(he)g (and)h(T)-7 b(emam)19 b([1993].)33 b(Here)19 b(again)f(w)n(e)i (consider)e(the)i(case)523 3260 y(when)29 b Fj(a)785 3272 y Fr(2)841 3260 y Fs(+)19 b Fj(a)969 3272 y Fr(3)1031 3260 y Fj(<)24 b Fs(0)29 b(and)f Fj(a)1397 3272 y Fr(2)1454 3260 y Fs(+)18 b Fj(ib)1602 3272 y Fr(2)1664 3260 y Fh(6)p Fs(=)24 b(0.)40 b(Let)29 b Fj(A)2070 3272 y Fr(0)2132 3260 y Fs(=)c Fh(\000)p Fj(a)2331 3272 y Fr(1)2396 3227 y Fi(@)2435 3202 y Fe(2)p 2377 3241 110 4 v 2377 3288 a Fi(@)t(x)2454 3272 y Fe(2)2525 3260 y Fs(and)j Fj(B)2750 3272 y Fr(0)2813 3260 y Fs(=)c Fh(\000)p Fj(ib)3032 3272 y Fr(1)3097 3227 y Fi(@)3136 3202 y Fe(2)p 3078 3241 V 3078 3288 a Fi(@)t(x)3155 3272 y Fe(2)3226 3260 y Fs(with)523 3359 y(p)r(erio)r(dic)33 b(b)r(oundary)e(conditions.)52 b(Since)33 b Fj(a)1936 3371 y Fr(1)2004 3359 y Fj(>)e Fs(0,)j Fj(A)2261 3371 y Fr(0)2331 3359 y Fs(is)f(a)f(self-adjoin)n(t)g (non-negativ)n(e)523 3459 y(de\014nite)26 b(op)r(erator)d(on)h Fh(D)r Fs(\()p Fj(A)1420 3471 y Fr(0)1459 3459 y Fs(\))f(=)g Fj(H)1678 3429 y Fr(2)1671 3480 y Fi(per)1798 3459 y Fs(and)i Fj(B)2020 3471 y Fr(0)2082 3459 y Fs(is)g(an)n(tisymmetric)f (with)i(resp)r(ect)e(to)h(the)523 3559 y(inner)i(pro)r(duct)h(\()p Fh(\001)p Fj(;)14 b Fh(\001)p Fs(\))28 b(and)g(the)g(semi-inner)f(pro)r (duct)g(\()p Fj(A)2334 3571 y Fr(0)2372 3559 y Fh(\001)p Fj(;)14 b Fh(\001)p Fs(\).)37 b(That)28 b(is)1571 3741 y(\()p Fj(B)1666 3753 y Fr(0)1703 3741 y Fj(u;)14 b(u)p Fs(\))23 b(=)f Fh(\000)p Fs(\()p Fj(u;)14 b(B)2223 3753 y Fr(0)2260 3741 y Fj(u)p Fs(\))1460 3924 y(\()p Fj(B)1555 3936 y Fr(0)1592 3924 y Fj(A)1654 3936 y Fr(0)1692 3924 y Fj(u;)g(u)p Fs(\))22 b(=)h Fh(\000)p Fs(\()p Fj(u;)14 b(B)2212 3936 y Fr(0)2248 3924 y Fj(A)2310 3936 y Fr(0)2348 3924 y Fj(u)p Fs(\))p Fj(:)523 4087 y Fs(Let)36 b Fj(N)747 4099 y Fr(0)784 4087 y Fs(\()p Fj(u)p Fs(\))h(=)g Fj(a)1079 4099 y Fr(0)1116 4087 y Fj(u)23 b Fs(+)h(\()p Fj(a)1352 4099 y Fr(2)1413 4087 y Fs(+)g Fj(ib)1567 4099 y Fr(2)1604 4087 y Fs(\))p Fh(j)p Fj(u)p Fh(j)1730 4057 y Fr(2)1767 4087 y Fj(u)f Fs(+)h(\()p Fj(a)2003 4099 y Fr(3)2064 4087 y Fs(+)g Fj(ib)2218 4099 y Fr(3)2254 4087 y Fs(\))p Fj(u)2348 4020 y Fg(R)2403 4040 y Fr(1)2387 4116 y(0)2454 4087 y Fh(j)p Fj(u)p Fh(j)2548 4057 y Fr(2)2585 4087 y Fj(dx)p Fs(,)39 b(then)e(\(1)p Fj(:)p Fs(1\))e(can)h(b)r(e)523 4186 y(written)28 b(as)1496 4286 y Fj(u)1544 4298 y Fi(t)1591 4286 y Fs(+)18 b(\()p Fj(A)1768 4298 y Fr(0)1825 4286 y Fs(+)g Fj(B)1971 4298 y Fr(0)2008 4286 y Fs(\))p Fj(u)23 b Fs(=)f Fj(N)2265 4298 y Fr(0)2302 4286 y Fs(\()p Fj(u)p Fs(\))803 b(\(5.1\))523 4435 y(Let)28 b(\002\()p Fj(r)r Fs(\))h(b)r(e)f(a)f(smo)r(oth)h(non-increasing)e(function)i(on)f([0)p Fj(;)14 b Fh(1)p Fs(])28 b(suc)n(h)f(that)h(\002\()p Fj(r)r Fs(\))d(=)e(1)k(for)523 4535 y Fj(r)f Fh(\024)c Fs(1)28 b(and)f(\002\()p Fj(r)r Fs(\))d(=)f(0)k(for)g Fj(r)f Fh(\025)c Fs(3,)28 b(and)f(\002)1849 4505 y Ff(0)1872 4535 y Fs(\()p Fj(r)r Fs(\))d Fh(\024)f Fs(1)k(for)g(all)h Fj(r)d Fh(2)f Fs([0)p Fj(;)14 b Fh(1)p Fs(].)36 b(Also)28 b(w)n(e)f(let)1465 4766 y(\011\()p Fj(u)p Fs(\))c(=)g(\002\()1860 4709 y Fh(k)p Fj(u)p Fh(k)1992 4721 y Fi(L)2038 4705 y Fc(1)p 1859 4746 241 4 v 1923 4822 a Fj(\032)1966 4834 y Ff(1)2110 4766 y Fs(\)\002\()2249 4709 y Fh(k)p Fj(u)p Fh(k)p 2249 4746 131 4 v 2275 4822 a Fj(\032)2318 4834 y Fr(0)2390 4766 y Fs(\))p Fj(:)1914 5147 y Fs(16)p eop %%Page: 17 17 17 16 bop 523 614 a Fs(Since)30 b(w)n(e)g(are)f(in)n(terested)g(in)h (the)h(long)e(time)h(b)r(eha)n(vior)f(of)h(the)g(solutions)f(then)i(w)n (e)e(will)523 714 y(concen)n(trate)f(on)h(the)g(b)r(eha)n(vior)f(at)h (the)h(absorbing)d(ball.)42 b(Therefore,)28 b(w)n(e)h(will)h(truncate) 523 814 y Fj(N)590 826 y Fr(0)627 814 y Fs(\()p Fj(u)p Fs(\))e(outside)f(the)h(absorbing)e(balls,)h(and)h(consider)e(the)i (prepared)f(equation)g(of)g(\(1)p Fj(:)p Fs(1\))1408 990 y Fj(u)1456 1002 y Fi(t)1503 990 y Fs(+)18 b(\()p Fj(A)1680 1002 y Fr(0)1736 990 y Fs(+)g Fj(B)1882 1002 y Fr(0)1919 990 y Fs(\))p Fj(u)23 b Fs(=)g(\011\()p Fj(u)p Fs(\))p Fj(N)2354 1002 y Fr(0)2391 990 y Fs(\()p Fj(u)p Fs(\))714 b(\(5.2\))523 1165 y(instead)27 b(of)h(equation)f(\(5)p Fj(:)p Fs(1\))g(itself.)38 b(Let)1604 1341 y Fj(R)q Fs(\()p Fj(u)p Fs(\))23 b(=)g(\011\()p Fj(u)p Fs(\))p Fj(N)2135 1353 y Fr(0)2171 1341 y Fs(\()p Fj(u)p Fs(\))p Fj(;)523 1517 y Fs(then)28 b(one)f(can)h(easily)e(v)n(erify)h(the)h(follo)n (wing)f(estimates)580 1726 y Fh(k)p Fj(R)q Fs(\()p Fj(u)p Fs(\))18 b Fh(\000)g Fj(R)q Fs(\()p Fj(v)s Fs(\))p Fh(k)83 b(\024)g Fs(\(7)p Fj(a)1461 1738 y Fr(0)1516 1726 y Fs(+)18 b(81)1683 1625 y Fg(q)p 1765 1625 321 4 v 1765 1726 a Fs(\()p Fj(a)1841 1697 y Fr(2)1841 1748 y(2)1897 1726 y Fs(+)g Fj(b)2016 1697 y Fr(2)2016 1748 y(2)2053 1726 y Fs(\))p Fj(\032)2128 1692 y Fr(2)2128 1746 y Ff(1)2217 1726 y Fs(+)g(13)2384 1625 y Fg(q)p 2466 1625 V 2466 1726 a Fs(\()p Fj(a)2542 1697 y Fr(2)2542 1748 y(3)2598 1726 y Fs(+)g Fj(b)2717 1697 y Fr(2)2717 1748 y(3)2754 1726 y Fs(\))p Fj(\032)2829 1692 y Fr(2)2829 1746 y(0)2866 1726 y Fs(\))p Fh(k)p Fj(w)r Fh(k)h Fs(+)f(\(3)p Fj(a)3263 1738 y Fr(0)1343 1912 y Fs(+27)1492 1811 y Fg(q)p 1573 1811 V 1573 1912 a Fs(\()p Fj(a)1649 1883 y Fr(2)1649 1934 y(2)1705 1912 y Fs(+)g Fj(b)1824 1883 y Fr(2)1824 1934 y(2)1861 1912 y Fs(\))p Fj(\032)1936 1878 y Fr(2)1936 1932 y Ff(1)2025 1912 y Fs(+)g(3)2150 1811 y Fg(q)p 2233 1811 V 101 x Fs(\()p Fj(a)2309 1883 y Fr(2)2309 1934 y(3)2365 1912 y Fs(+)g Fj(b)2484 1883 y Fr(2)2484 1934 y(3)2520 1912 y Fs(\))q Fj(\032)2596 1878 y Fr(2)2596 1932 y(0)2633 1912 y Fs(\))p Fh(k)p Fj(w)r Fh(k)2810 1878 y Fr(1)p Fi(=)p Fr(2)2914 1912 y Fh(k)p Fj(A)3018 1878 y Fr(1)p Fi(=)p Fr(2)3122 1912 y Fj(w)r Fh(k)3225 1878 y Fr(1)p Fi(=)p Fr(2)1195 2068 y Fh(\024)83 b Fj(\026)1393 2080 y Fr(1)1430 2068 y Fh(k)p Fj(w)r Fh(k)18 b Fs(+)g Fj(\026)1726 2080 y Fr(1)1763 2068 y Fh(k)p Fj(w)r Fh(k)1908 2034 y Fr(1)p Fi(=)p Fr(2)2013 2068 y Fh(k)p Fj(A)2117 2034 y Fr(1)p Fi(=)p Fr(2)2221 2068 y Fj(w)r Fh(k)2324 2034 y Fr(1)p Fi(=)p Fr(2)3217 2068 y Fs(\(5.3\))523 2244 y(where)27 b Fj(w)f Fs(=)c Fj(u)c Fh(\000)g Fj(v)31 b Fs(and)d Fj(\026)1367 2256 y Fr(1)1427 2244 y Fs(=)23 b(\(7)p Fj(a)1633 2256 y Fr(0)1688 2244 y Fs(+)18 b(81)1855 2172 y Fg(p)p 1937 2172 V 1937 2244 a Fs(\()p Fj(a)2013 2215 y Fr(2)2013 2266 y(2)2069 2244 y Fs(+)g Fj(b)2188 2215 y Fr(2)2188 2266 y(2)2225 2244 y Fs(\))p Fj(\032)2300 2214 y Fr(2)2300 2264 y Ff(1)2389 2244 y Fs(+)g(13)2556 2172 y Fg(p)p 2638 2172 V 2638 2244 a Fs(\()p Fj(a)2714 2215 y Fr(2)2714 2266 y(3)2770 2244 y Fs(+)g Fj(b)2889 2215 y Fr(2)2889 2266 y(3)2926 2244 y Fs(\))p Fj(\032)3001 2214 y Fr(2)3001 2264 y(0)3038 2244 y Fs(\).)648 2343 y(W)-7 b(e)31 b(call)f(a)g(n)n(um)n(b)r(er)g Fj(\025)f Fh(2)f Fs(\(0)p Fj(;)14 b Fh(1)p Fs(\))31 b(a)f Fn(sp)l(e)l(ctr)l(al)j (b)l(arrier)e Fs(for)g(the)f(equation)g(\(1)p Fj(:)p Fs(1\))h(if)g(for)523 2443 y(ev)n(ery)26 b(t)n(w)n(o)h(distinct)h (solutions)f Fj(u)p Fs(,)h Fj(v)e Fh(2)d(D)r Fs(\()p Fj(A)p Fs(\))29 b(satisfying)1401 2619 y(\()p Fj(A)p Fs(\()p Fj(u)19 b Fh(\000)f Fj(v)s Fs(\))p Fj(;)c(u)19 b Fh(\000)f Fj(v)s Fs(\))23 b(=)g Fj(\025)p Fh(k)p Fj(u)18 b Fh(\000)g Fj(v)s Fh(k)2449 2585 y Fr(2)2486 2619 y Fj(;)523 2795 y Fs(it)28 b(follo)n(ws)f(that)948 2971 y Fh(k)p Fs(\()p Fj(A)19 b Fh(\000)f Fj(\025)p Fs(\)\()p Fj(u)h Fh(\000)f Fj(v)s Fs(\))p Fh(k)1565 2937 y Fr(2)1620 2971 y Fs(+)g(\()p Fj(R)q Fs(\()p Fj(u)p Fs(\))h Fh(\000)f Fj(R)q Fs(\()p Fj(v)s Fs(\))p Fj(;)c Fs(\()p Fj(A)19 b Fh(\000)f Fj(\025)p Fs(\)\()p Fj(u)h Fh(\000)f Fj(v)s Fs(\)\))24 b Fj(>)f Fs(0)p Fj(:)523 3147 y Fs(W)-7 b(e)29 b(remark)d(here)i(that)g(a)g(sp)r(ectral)f(barrier)g(cannot)g(b)r(e)i (an)f(eigen)n(v)-5 b(alue)27 b(of)h Fj(A)p Fs(.)39 b(F)-7 b(urther-)523 3246 y(more,)39 b(it)e(follo)n(ws)f(from)h(\(5)p Fj(:)p Fs(3\))g(that)g(in)g(order)f(for)g Fj(\025)i Fs(to)f(b)r(e)g(a)g (sp)r(ectral)f(barrier,)i(it)f(is)523 3346 y(su\016cien)n(t)28 b(that)1326 3446 y Fh(k)p Fs(\()p Fj(A)19 b Fh(\000)f Fj(\025)p Fs(\)\()p Fj(w)r Fs(\))p Fh(k)24 b(\025)f Fs(\()p Fj(\026)2005 3458 y Fr(1)2061 3446 y Fs(+)18 b Fj(\026)2194 3458 y Fr(1)2231 3446 y Fj(\025)2279 3411 y Fr(1)p Fi(=)p Fr(4)2384 3446 y Fs(\))p Fh(k)p Fj(w)r Fh(k)p Fj(:)633 b Fs(\(5.4\))523 3591 y(So)28 b(if)g(w)n(e)f(let)i Fj(\025)f Fs(b)r(e)g(the)g(midp)r(oin)n(t)1656 3558 y Fr(1)p 1656 3572 34 4 v 1656 3620 a(2)1699 3591 y Fs(\()p Fj(\025)1779 3603 y Fi(N)6 b Fr(+1)1945 3591 y Fs(+)18 b Fj(\025)2076 3603 y Fi(N)2140 3591 y Fs(\))28 b(of)g(t)n(w)n(o)f(consecutiv)n(e)g (eigen)n(v)-5 b(alues)26 b(of)523 3691 y Fj(A)p Fs(,)i(then)1306 3825 y Fh(k)p Fs(\()p Fj(A)18 b Fh(\000)g Fj(\025)p Fs(\)\()p Fj(w)r Fs(\))p Fh(k)25 b(\025)1912 3769 y Fs(1)p 1912 3806 42 4 v 1912 3882 a(2)1964 3825 y(\()p Fj(\025)2044 3837 y Fi(N)6 b Fr(+1)2210 3825 y Fh(\000)18 b Fj(\025)2341 3837 y Fi(N)2405 3825 y Fs(\))p Fh(k)p Fj(w)r Fh(k)p Fj(;)523 3998 y Fs(and)27 b(hence)h(equation)f(\(5)p Fj(:)p Fs(4\))g(is)h(satis\014ed)f(if)1384 4157 y(1)p 1384 4194 V 1384 4270 a(2)1436 4213 y(\()p Fj(\025)1516 4225 y Fi(N)6 b Fr(+1)1682 4213 y Fh(\000)18 b Fj(\025)1813 4225 y Fi(N)1876 4213 y Fs(\))24 b Fh(\025)e Fs(\()p Fj(\026)2101 4225 y Fr(1)2157 4213 y Fs(+)c Fj(\026)2290 4225 y Fr(1)2328 4213 y Fj(\025)2376 4179 y Fr(1)p Fi(=)p Fr(4)2481 4213 y Fs(\))p Fj(:)523 4412 y Fs(In)28 b(other)f(w)n(ords,) 1514 4546 y Fj(N)h Fs(+)1701 4490 y(1)p 1701 4527 V 1701 4603 a(2)1776 4546 y Fh(\025)1895 4490 y Fj(\026)1945 4502 y Fr(1)p 1874 4527 130 4 v 1874 4603 a Fs(4)p Fj(\031)1966 4579 y Fr(2)2013 4546 y Fs(\(1)18 b(+)g Fj(\025)2236 4512 y Fr(1)p Fi(=)p Fr(4)2341 4546 y Fs(\))p Fj(:)821 b Fs(\(5.5\))648 4715 y(On)27 b(the)h(other)f(hand,)h(the)g(existence)f (of)g(inertial)h(manifolds)f(also)f(requires)h(that)1546 4891 y Fj(\025)d(>)e Fs(2)14 b(max)o Fh(f)p Fj(\025)2005 4903 y Fr(0)p Fi(;)p Fr(1)2095 4891 y Fj(;)g(\025)2180 4912 y Fr(0)p Fi(;)2243 4889 y Fe(3)p 2243 4898 29 4 v 2243 4932 a(4)2286 4891 y Fh(g)p Fj(;)866 b Fs(\(5.6\))1914 5147 y(17)p eop %%Page: 18 18 18 17 bop 523 614 a Fs(for)27 b Fj(\025)698 626 y Fr(0)p Fi(;)p Fr(1)789 614 y Fj(;)14 b(\025)874 635 y Fr(0)p Fi(;)937 613 y Fe(3)p 937 622 29 4 v 937 655 a(4)1007 614 y Fs(satisfying)523 827 y(\()p Fj(R)q Fs(\()p Fj(u)p Fs(\))19 b Fh(\000)f Fj(R)q Fs(\()p Fj(v)s Fs(\))p Fj(;)c Fs(\()p Fj(u)k Fh(\000)g Fj(v)s Fs(\)\))24 b Fh(\025)f(\000)p Fj(\025)1554 839 y Fr(0)p Fi(;)p Fr(1)1644 827 y Fh(k)p Fj(u)17 b Fh(\000)h Fj(v)s Fh(k)1919 793 y Fr(2)1975 827 y Fh(\000)g Fj(\025)2106 848 y Fr(0)p Fi(;)2169 825 y Fe(3)p 2169 834 V 2169 868 a(4)2211 827 y Fs(\()p Fh(k)p Fj(u)g Fh(\000)g Fj(v)s Fh(k)2529 771 y Fe(3)p 2529 780 V 2529 813 a(4)2599 827 y Fh(k)p Fj(A)2703 793 y Fr(1)p Fi(=)p Fr(2)2807 827 y Fs(\()p Fj(u)g Fh(\000)g Fj(v)s Fs(\))p Fh(k)3105 793 y Fr(\(1)p Ff(\000)3226 771 y Fe(3)p 3226 780 V 3226 813 a(4)3264 793 y Fr(\))3295 827 y Fs(\))3327 793 y Fr(2)3364 827 y Fj(:)523 1019 y Fs(In)28 b(our)f(case,)f(it)i (follo)n(ws)f(from)g(equation)g(\(5)p Fj(:)p Fs(3\))h(that)1416 1215 y Fj(\025)1464 1227 y Fr(0)p Fi(;)p Fr(1)1578 1215 y Fs(=)22 b Fj(\026)1715 1227 y Fr(1)1836 1215 y Fs(and)82 b Fj(\025)2100 1236 y Fr(0)p Fi(;)2164 1213 y Fe(3)p 2164 1222 V 2164 1256 a(4)2229 1215 y Fs(=)23 b Fj(\026)2367 1172 y Fr(4)p Fi(=)p Fr(3)2367 1237 y(1)2471 1215 y Fj(:)723 b Fs(\(5.7\))523 1407 y(So)27 b(equation)g(\(5)p Fj(:)p Fs(6\))h(is)f(equiv)-5 b(alen)n(t)28 b(to)1578 1603 y Fj(\025)c(>)e Fs(2)14 b(max)o Fh(f)p Fj(\026)2039 1615 y Fr(1)2076 1603 y Fj(;)g(\026)2163 1560 y Fr(4)p Fi(=)p Fr(3)2163 1625 y(1)2267 1603 y Fh(g)p Fj(:)885 b Fs(\(5.8\))648 1784 y(Therefore)22 b(w)n(e)i(ha)n(v)n(e)f(from)g(\(5)p Fj(:)p Fs(5\))h(and)g(\(5)p Fj(:)p Fs(8\))f(that)i(there)f(exists)f(an) h(inertial)f(manifold)523 1883 y Fh(M)k Fs(whose)g(dimension)h Fj(dim)p Fs(\()p Fh(M)p Fs(\))f(satis\014es)g(:)1097 2103 y Fj(dim)p Fs(\()p Fh(M)p Fs(\))18 b(+)1517 2046 y(1)p 1517 2083 42 4 v 1517 2160 a(2)1592 2103 y Fj(>)1710 2046 y(\026)1760 2058 y Fr(1)p 1690 2083 130 4 v 1690 2160 a Fs(4)p Fj(\031)1782 2136 y Fr(2)1829 2103 y Fs(\(1)g(+)g(\(2)c (max)o Fh(f)p Fj(\026)2338 2115 y Fr(1)2375 2103 y Fj(;)g(\026)2462 2059 y Fr(4)p Fi(=)p Fr(3)2462 2125 y(1)2566 2103 y Fh(g)p Fs(\))2640 2068 y Fr(1)p Fi(=)p Fr(4)2744 2103 y Fs(\))p Fj(;)418 b Fs(\(5.9\))523 2324 y(where)27 b(for)g Fh(j)p Fj(b)949 2336 y Fr(2)986 2324 y Fh(j)c(\024)1120 2255 y(p)p 1189 2255 42 4 v 69 x Fs(3)p Fh(j)p Fj(a)1298 2336 y Fr(2)1335 2324 y Fh(j)p Fs(,)1023 2563 y Fj(\026)1073 2575 y Fr(1)1193 2563 y Fs(=)83 b(\(7)18 b(+)g(81)1610 2507 y(\()p Fj(a)1686 2477 y Fr(2)1686 2528 y(2)1741 2507 y Fs(+)g Fj(b)1860 2477 y Fr(2)1860 2528 y(2)1897 2507 y Fs(\))1929 2477 y Fr(1)p Fi(=)p Fr(2)p 1609 2544 425 4 v 1666 2620 a Fh(j)p Fj(a)1733 2632 y Fr(2)1789 2620 y Fs(+)g Fj(a)1916 2632 y Fr(3)1953 2620 y Fh(j)2062 2563 y Fs(+)g(13)2239 2507 y(\()p Fj(a)2315 2477 y Fr(2)2315 2528 y(3)2370 2507 y Fs(+)g Fj(b)2489 2477 y Fr(2)2489 2528 y(3)2526 2507 y Fs(\))2558 2477 y Fr(1)p Fi(=)p Fr(2)p 2238 2544 V 2295 2620 a Fh(j)p Fj(a)2362 2632 y Fr(2)2418 2620 y Fs(+)g Fj(a)2545 2632 y Fr(3)2582 2620 y Fh(j)2672 2563 y Fs(\))p Fj(a)2748 2575 y Fr(0)2804 2563 y Fs(+)1341 2826 y(162)1477 2770 y(\()p Fj(a)1553 2740 y Fr(2)1553 2790 y(2)1607 2770 y Fs(+)g Fj(b)1726 2740 y Fr(2)1726 2790 y(2)1763 2770 y Fs(\))1795 2740 y Fr(1)p Fi(=)p Fr(2)p 1475 2807 V 1533 2883 a Fh(j)p Fj(a)1600 2895 y Fr(2)1655 2883 y Fs(+)g Fj(a)1782 2895 y Fr(3)1820 2883 y Fh(j)1910 2681 y Fg(s)p 1993 2681 639 4 v 2023 2770 a Fs(1)p 2003 2807 82 4 v 2003 2883 a Fj(a)2047 2895 y Fr(1)2094 2826 y Fs(\(1)g(+)2393 2770 y Fj(a)2437 2782 y Fr(3)p 2279 2807 310 4 v 2279 2883 a Fh(j)p Fj(a)2346 2895 y Fr(2)2402 2883 y Fs(+)g Fj(a)2529 2895 y Fr(3)2566 2883 y Fh(j)2599 2826 y Fs(\))28 b Fj(a)2703 2783 y Fr(3)p Fi(=)p Fr(2)2703 2848 y(0)2807 2826 y Fj(;)345 b Fs(\(5.10\))523 3091 y(and)27 b(for)h Fh(j)p Fj(b)871 3103 y Fr(2)907 3091 y Fh(j)c Fj(>)1041 3022 y Fh(p)p 1110 3022 42 4 v 69 x Fs(3)p Fh(j)p Fj(a)1219 3103 y Fr(2)1256 3091 y Fh(j)p Fs(,)668 3330 y Fj(\026)718 3342 y Fr(1)838 3330 y Fs(=)83 b(\(7)18 b(+)g(81)1255 3274 y(\()p Fj(a)1331 3244 y Fr(2)1331 3295 y(2)1386 3274 y Fs(+)g Fj(b)1505 3244 y Fr(2)1505 3295 y(2)1542 3274 y Fs(\))1574 3244 y Fr(1)p Fi(=)p Fr(2)p 1254 3311 425 4 v 1311 3387 a Fh(j)p Fj(a)1378 3399 y Fr(2)1434 3387 y Fs(+)g Fj(a)1561 3399 y Fr(3)1598 3387 y Fh(j)1707 3330 y Fs(+)g(13)1884 3274 y(\()p Fj(a)1960 3244 y Fr(2)1960 3295 y(3)2015 3274 y Fs(+)g Fj(b)2134 3244 y Fr(2)2134 3295 y(3)2171 3274 y Fs(\))2203 3244 y Fr(1)p Fi(=)p Fr(2)p 1883 3311 V 1940 3387 a Fh(j)p Fj(a)2007 3399 y Fr(2)2063 3387 y Fs(+)g Fj(a)2190 3399 y Fr(3)2227 3387 y Fh(j)2317 3330 y Fs(\))p Fj(a)2393 3342 y Fr(0)2449 3330 y Fs(+)g(162)p Fj(\016)2708 3274 y Fs(\()p Fj(a)2784 3244 y Fr(2)2784 3295 y(2)2838 3274 y Fs(+)h Fj(b)2958 3244 y Fr(2)2958 3295 y(2)2994 3274 y Fs(\))3026 3244 y Fr(1)p Fi(=)p Fr(2)p 2707 3311 V 2745 3387 a Fh(j)p Fj(a)2812 3399 y Fr(2)2868 3387 y Fs(+)f Fj(a)2995 3399 y Fr(3)3032 3387 y Fh(j)3055 3363 y Fr(2)3159 3330 y Fh(\002)986 3602 y Fs([1)g(+)1152 3448 y Fg(s)p 1235 3448 1253 4 v 154 x Fs(1)g(+)1388 3545 y Fj(a)1432 3557 y Fr(1)1469 3545 y Fh(j)p Fj(a)1536 3557 y Fr(2)1592 3545 y Fs(+)g Fj(a)1719 3557 y Fr(3)1756 3545 y Fh(j)p Fs(\()p Fh(j)p Fj(a)1878 3557 y Fr(2)1934 3545 y Fs(+)g Fj(a)2061 3557 y Fr(3)2098 3545 y Fh(j)g Fs(+)g Fj(\016)k Fs(+)c Fj(a)2408 3557 y Fr(3)2445 3545 y Fs(\))p 1388 3582 1090 4 v 1853 3659 a Fj(\016)1893 3635 y Fr(2)1931 3659 y Fj(a)1975 3671 y Fr(0)2515 3602 y Fs(])p Fj(a)2582 3567 y Fr(2)2582 3622 y(0)2619 3602 y Fj(:)533 b Fs(\(5.11\))648 3840 y(In)27 b(conclusion)g(w)n(e)g(ha)n(v)n(e)g(the)h(follo)n(wing)e (theorem:)523 4021 y Fo(Theorem)k(3)42 b Fn(L)l(et)34 b Fj(a)1210 4033 y Fr(1)1279 4021 y Fj(>)d Fs(0)p Fj(;)14 b(a)1498 4033 y Fr(2)1566 4021 y Fh(\024)32 b Fs(0)i Fn(and)h Fj(a)1949 4033 y Fr(2)2008 4021 y Fs(+)22 b Fj(a)2139 4033 y Fr(3)2208 4021 y Fj(<)31 b Fs(0)p Fn(.)53 b(Then)35 b(ther)l(e)f(exists)h(an)f(iner-)523 4120 y(tial)29 b(manifold)h Fh(M)e Fn(for)i(the)e(dynamic)l(al)i(system)f (\(1.1\)-\(1.2\)-\(1.3\))i(whose)e(dimension)h(is)523 4220 y(sp)l(e)l(ci\014e)l(d)h(in)e(\(5.9\)-\(5.10\)-\(5.11\).)648 4400 y Fs(W)-7 b(e)28 b(can)g(see)f(the)h(relation)g(of)f(the)i(upp)r (er)f(b)r(ound)g(of)g(the)h(dimension)e(of)h(the)h(inertial)523 4500 y(manifold)38 b(for)f(equation)g(\(1.1\))h(with)g(that)g(for)f (the)h(corresp)r(onding)e(cubic)i(Ginzburg-)523 4599 y(Landau)23 b(equation)h(\(1.4\),)g(b)n(y)g(considering)f(v)-5 b(arious)23 b(v)-5 b(alues)23 b(of)h Fj(a)2576 4611 y Fr(3)2613 4599 y Fs(.)36 b(Primarily)-7 b(,)23 b(if)i Fj(a)3175 4611 y Fr(3)3235 4599 y Fj(<)e Fs(0,)523 4699 y(then)31 b(it)g(follo)n(ws)e(from)h(\(5)p Fj(:)p Fs(10\))g(and)g(\(5)p Fj(:)p Fs(11\))f(that)i(the)g(dimension)f(of)g(the)h(inertial)f(mani-) 523 4799 y(fold)f(for)f(the)h(nonlo)r(cal)f(amplitude)h(equation)f (will)g(b)r(e)h(smaller)f(than)h(that)g(of)f(the)h(cubic)523 4898 y(Ginzburg-Landau)37 b(equation.)70 b(This)39 b(is)g(actually)f (what)h(w)n(e)f(w)n(ould)h(exp)r(ect)g(b)r(ecause)1914 5147 y(18)p eop %%Page: 19 19 19 18 bop 523 614 a Fs(the)35 b(nonlo)r(cal)f(term)h(no)n(w)f(b)r(eha)n (v)n(es)f(lik)n(e)i(a)f(stabilizing)g(term.)58 b(On)35 b(the)g(other)f(hand)h(if)523 714 y(0)27 b Fj(<)g(a)728 726 y Fr(3)792 714 y Fj(<)g Fh(j)p Fj(a)951 726 y Fr(2)989 714 y Fh(j)p Fs(,)k(then)f(the)h(upp)r(er)f(b)r(ound)h(of)f(the)h (dimension)f(of)g(the)h(inertial)e(manifold)523 814 y(will)36 b(b)r(e)h(larger.)61 b(That)36 b(is,)i(due)f(to)f(the)g(no)n (w-defectiv)n(e)f(\(or)h(no)n(w-destabilizing\))f(non-)523 913 y(lo)r(cal)f(term,)i(and)f(the)g(long)f(time)h(dynamics)f(of)h(the) f(equation)g(\(1.1\))h(is)f(describ)r(ed)h(b)n(y)523 1013 y(a)f(higher)h(dimensional)f(ordinary)f(di\013eren)n(tial)i (systems.)58 b(Moreo)n(v)n(er)32 b(as)i Fj(a)2977 1025 y Fr(3)3084 1013 y Fh(\045)70 b(j)p Fj(a)3304 1025 y Fr(2)3341 1013 y Fh(j)p Fs(,)523 1112 y(the)30 b(upp)r(er)f(b)r(ound)h (of)f(the)h(dimension)g(of)f(the)h(inertial)f(manifold)g(will)h(get)f (enormously)523 1212 y(large.)39 b(This)29 b(is)g(ob)n(viously)e(true)i (b)r(ecause)f(when)h Fj(a)2153 1224 y Fr(2)2216 1212 y Fs(=)24 b Fh(\000)p Fj(a)2414 1224 y Fr(3)2480 1212 y Fs(the)29 b(equation)f(is)h(no)g(longer)523 1312 y(dissipativ)n(e.)39 b(Indeed,)29 b(as)f(it)h(is)g(clearly)e(seen)i(from)f(equation)g (\(2.3\))h(in)f(this)h(case)f(w)n(e)g(are)523 1411 y(no)j(longer)f (guaran)n(teed)g(the)i(existence)f(of)g(an)h(absorbing)d(ball.)49 b(Our)30 b(last)i(emphasis)e(is)523 1511 y(that)25 b(these)g(estimates) f(on)h(the)g(dimesions)f(dep)r(end)i(the)f(sizes)f(of)h(the)g (absorbing)e(balls)h(so)523 1611 y(when)k(the)g(sizes)f(of)h(the)g (absorbing)e(balls)h(increase/decrease,)d(not)k(only)f(the)h(diameters) 523 1710 y(of)c(the)h(inertial)f(manifolds)g(or)f(the)i(global)e (attractor)g(get)h(larger/smaller)c(but)25 b(also)e(their)523 1810 y(dimensions)k(get)h(larger/smaller)c(as)j(w)n(ell.)523 2084 y Fk(6)135 b(Determining)45 b(no)t(des)e(and)g(other)h(degrees)g (of)g(free-)725 2234 y(dom)523 2416 y Fs(A)23 b(\014nite)f(set)h(of)f (p)r(oin)n(ts)g Fh(E)30 b Fs(=)23 b Fh(f)p Fj(x)1523 2428 y Fr(1)1560 2416 y Fj(;)14 b(x)1644 2428 y Fr(2)1682 2416 y Fj(;)g Fh(\001)g(\001)g(\001)f Fj(;)h(x)1913 2428 y Fi(N)1976 2416 y Fh(g)23 b(\032)g Fs([0)p Fj(;)14 b(l)r Fs(])21 b(is)h(called)g(determining)g(if)g(for)g(an)n(y)523 2515 y(t)n(w)n(o)27 b(solutions)f Fj(u;)14 b(v)30 b Fs(of)e(\(1.1\))f (for)g(whic)n(h)g(max)1993 2527 y Fr(1)p Ff(\024)p Fi(i)p Ff(\024)p Fi(N)2230 2515 y Fh(j)p Fj(u)p Fs(\()p Fj(x)2380 2527 y Fi(i)2408 2515 y Fj(;)14 b(t)p Fs(\))k Fh(\000)f Fj(v)s Fs(\()p Fj(x)2729 2527 y Fi(i)2758 2515 y Fj(;)d(t)p Fs(\))p Fh(j)23 b(!)g Fs(0,)28 b(w)n(e)f(ha)n(v)n(e)523 2615 y Fh(k)p Fj(u)6 b Fh(\000)g Fj(v)s Fh(k)22 b(!)h Fs(0.)34 b(This)21 b(notion)g(of)h(degrees)e(of)h(freedom)g(w)n(as)g (\014rst)g(in)n(tro)r(duced)g(b)n(y)g(F)-7 b(oias)21 b(and)523 2715 y(T)-7 b(emam)30 b([1984])f(in)i(the)g(con)n(text)f(of)h (the)f(Na)n(vier-Stok)n(es)f(equations.)44 b(F)-7 b(or)30 b(estimates)h(to)523 2814 y(the)c(n)n(um)n(b)r(er)g(of)g(determining)g (no)r(des)g(and)g(other)f(kinds)h(of)g(degrees)f(of)h(freedom)f(for)h (the)523 2914 y(Na)n(vier-Stok)n(es)32 b(equations)h(see)h(Jones)f(and) h(Titi)g([1992,1993],)e(Co)r(c)n(kburn,)j(Jones)f(and)523 3013 y(Titi)d([1995])e(and)h(references)f(therein.)45 b(Kuk)-5 b(a)n(vica)29 b([1992])g(has)h(sho)n(wn)f(that)i(for)f(the)h (cu-)523 3113 y(bic)d(complex)g(Ginzburg-Landau)e(equation)i(\(1.4\))g (there)f(are)g(t)n(w)n(o)h(determining)g(no)r(des.)523 3213 y(The)k(pro)r(of)e(is)i(based)f(on)g(the)h(fact)f(that)h(w)n(e)f (ha)n(v)n(e)g(one)g(spatial)f(v)-5 b(ariable)31 b(and)g(that)h(the)523 3312 y(solutions)25 b(of)h(equation)f(\(1.4\))g(b)r(elong)h(to)f(a)g (Gevrey)g(class)g(of)h(regularit)n(y)-7 b(,)24 b(and)i(hence)f(are)523 3412 y(real)k(analytic.)44 b(Let)30 b(us)g(observ)n(e)e(again)h(that)i (follo)n(wing)e(the)h(w)n(ork)f(of)h(F)-7 b(errari)28 b(and)i(Titi)523 3512 y([1994],)22 b(whic)n(h)g(is)h(based)f(on)g(the)h (w)n(ork)e(of)h(F)-7 b(oias)22 b(and)g(T)-7 b(emam)22 b([1989],)g(w)n(e)g(can)g(sho)n(w)g(that)523 3611 y(the)33 b(solutions)e(of)h(the)h(nonlo)r(cal)e(complex)h(Ginzburg-Landau)e (equation)i(\(1.1\))g(b)r(elong)523 3711 y(to)c(a)f(Gevrey)g(class)g (of)h(regularit)n(y)-7 b(,)26 b(and)i(are)f(real)g(analytic.)37 b(Then,)28 b(follo)n(wing)f(the)h(w)n(ork)523 3810 y(of)e(Kuk)-5 b(a)n(vica)24 b([1992])f(w)n(e)j(can)f(sho)n(w)f(that)i(equation)f (\(1.1\))g(has)g(t)n(w)n(o)g(determining)h(no)r(des.)523 3910 y(In)j(Section)f(5)g(it)h(w)n(as)f(sho)n(wn)f(that)i(the)g (inertial)f(manifolds)g(are)g(parameterized)f(b)n(y)h(the)523 4010 y(F)-7 b(ourier)23 b(mo)r(des.)35 b(In)24 b(F)-7 b(oias)23 b(and)g(Titi)h([1991])e(it)i(w)n(as)f(sho)n(wn)f(that)i(the)g (inertial)g(manifolds)523 4109 y(for)32 b(man)n(y)f(equations,)i(suc)n (h)f(as)g(Kuramoto-Siv)-5 b(ashinsky)29 b(equation)j(and)g(the)h (complex)523 4209 y(Ginzburg-Landau)c(equation)h(\(1.1\),)i(can)e(b)r (e)h(parameterized)e(b)n(y)i(the)g(no)r(dal)f(v)-5 b(alues)30 b(of)523 4309 y(the)i(solutions)f(or)f(b)n(y)h(the)h(lo)r(cal)f(v)n (olume)g(elemen)n(ts.)48 b(Moreo)n(v)n(er,)30 b(it)i(is)g(sho)n(wn)e (that)i(the)523 4408 y(n)n(um)n(b)r(er)20 b(of)g(no)r(des)h(or)e(v)n (olume)h(elemen)n(ts)g(needed)h(for)e(suc)n(h)h(parameterization)f(is)h (prop)r(or-)523 4508 y(tional)28 b(to)g(the)g(dimension)h(of)f(the)g (inertial)g(manifold.)39 b(This)28 b(kind)g(of)g(parameterization)523 4607 y(w)n(as)33 b(recen)n(tly)h(generalized)f(b)n(y)h(Co)r(c)n(kburn,) h(Jones)e(and)i(Titi)f([1995])f(to)h(include)h(\014nite)523 4707 y(elemen)n(ts)h(and)h(other)f(kinds)g(of)h(in)n(terp)r(olan)n(t)f (p)r(olynomials.)63 b(W)-7 b(e)37 b(observ)n(e)d(that)j(these)523 4807 y(results)25 b(are)g(v)-5 b(alid)26 b(for)g(the)g(nonlo)r(cal)f (complex)h(Ginzburg-Landau)f(equation)g(\(1.1\).)36 b(W)-7 b(e)1914 5147 y(19)p eop %%Page: 20 20 20 19 bop 523 614 a Fs(omit)27 b(the)h(details)e(b)r(ecause)h(they)g (are)f(w)n(ord)g(b)n(y)h(w)n(ord)f(as)g(in)i(F)-7 b(oias)26 b(and)h(Titi)g([1991])f(and)523 714 y(Co)r(c)n(kburn,)h(Jones)f(and)i (Titi)g([1995].)523 983 y Fk(7)135 b(The)44 b(case)i Fb(a)1345 1001 y Fs(2)1424 983 y Fa(=)33 b Fb(b)1598 1001 y Fs(2)1677 983 y Fa(=)g(0)523 1165 y Fs(In)28 b(this)g(section)f (w)n(e)g(consider)g(equation)g(\(1.1\))g(with)h Fj(a)2282 1177 y Fr(2)2342 1165 y Fs(=)23 b Fj(b)2466 1177 y Fr(2)2526 1165 y Fs(=)g(0,)k(i.e.)1032 1378 y Fj(u)1080 1390 y Fi(t)1132 1378 y Fs(=)22 b Fj(a)1263 1390 y Fr(0)1300 1378 y Fj(u)c Fs(+)g(\()p Fj(a)1525 1390 y Fr(1)1581 1378 y Fs(+)g Fj(ib)1729 1390 y Fr(1)1766 1378 y Fs(\))p Fj(u)1846 1390 y Fi(xx)1943 1378 y Fs(+)g(\()p Fj(a)2102 1390 y Fr(3)2158 1378 y Fs(+)g Fj(ib)2306 1390 y Fr(3)2343 1378 y Fs(\))p Fj(u)2433 1322 y Fs(1)p 2433 1359 42 4 v 2441 1435 a Fj(l)2498 1265 y Fg(Z)2581 1286 y Fi(l)2544 1454 y Fr(0)2620 1378 y Fh(j)p Fj(u)p Fh(j)2714 1344 y Fr(2)2751 1378 y Fj(dx;)353 b Fs(\(7.1\))523 1571 y(sub)5 b(ject)29 b(to)g(the)g(same)g(p)r(erio)r(dic)g(b)r(oundary)f (condition.)41 b(Again)28 b(w)n(e)h(rescale)f(the)h(spatial)523 1671 y(v)-5 b(ariable)18 b(to)g(obtain)g(that)h(the)g(p)r(erio)r(dic)f (in)n(terv)-5 b(al)18 b(length)h Fj(l)24 b Fs(=)f(1.)33 b(F)-7 b(or)18 b(the)h(global)f(existence)523 1771 y(it)31 b(is)g(no)n(w)f(required)g(that)h Fj(a)1428 1783 y Fr(3)1493 1771 y Fh(\024)d Fs(0;)k(otherwise)e(as)g(sho)n(wn)g(in)h(section)g(2)f (for)g(su\016cien)n(tly)523 1870 y(large)21 b(initial)i(data)f(the)h (solution)f(will)h(blo)n(w)f(up)h(in)g(\014nite)g(time.)36 b(In)23 b(addition,)g(in)g(the)g(case)523 1970 y(of)30 b Fj(a)664 1982 y Fr(3)728 1970 y Fj(<)c Fs(0)j(one)h(can)f(follo)n(w)g (the)h(argumen)n(t)f(in)h(Section)g(3)f(to)h(obtain)f(estimates)h(on)f (the)523 2070 y(sizes)h(of)g(the)g(absorbing)f(balls)h(in)g Fj(L)1693 2082 y Fr(2)1730 2070 y Fs(,)h Fj(L)1841 2082 y Fr(2)1908 2070 y Fs(of)f Fj(u)2053 2082 y Fi(x)2095 2070 y Fs(,)h(and)f Fj(L)2370 2039 y Ff(1)2470 2070 y Fs(norms)f(for)h(equation)g(\(7.1\).)523 2169 y(Ho)n(w)n(ev)n(er,)h (this)h(is)f(a)g(simple)h(case)f(in)h(whic)n(h)f(w)n(e)h(will)f(b)r(e)h (able)g(to)f(construct)g(explicitly)523 2269 y(the)h(Inertial)e (Manifold)h(of)g(dimension)g(equals)g(to)g(that)g(of)g(the)h(global)e (attractor.)45 b(The)523 2368 y(details)30 b(here)g(are)f(in)h(the)h (spirit)f(of)g(the)g(w)n(ork)f(of)h(Blo)r(c)n(h)g(and)g(Titi)g([1990].) 43 b(Notice)30 b(that)523 2468 y Fh(f)p Fj(e)604 2438 y Fr(2)p Fi(\031)r(k)q(xi)779 2468 y Fh(g)821 2480 y Fi(k)q Ff(2)p Fi(Z)986 2468 y Fs(is)f(a)h(complete)g(set)g(in)g Fj(L)1784 2438 y Fr(2)1784 2489 y Fi(per)1916 2468 y Fs(of)g(eigenfunctions)g(of)f(the)i(op)r(erator)d Fj(A)p Fs(.)44 b(Let)523 2579 y Fj(H)592 2591 y Fi(m)687 2579 y Fs(=)33 b(span)p Fh(f)p Fj(e)1033 2549 y Fr(2)p Fi(\031)r(k)q(xi)1273 2579 y Fs(:)65 b Fh(\000)p Fj(m)32 b Fh(\024)f Fj(k)k Fh(\024)d Fj(m)p Fh(g)h Fs(and)f Fj(P)2169 2591 y Fi(m)2265 2579 y Fs(:)g Fj(L)2377 2549 y Fr(2)2377 2599 y Fi(per)2479 2579 y Fs(\(0)p Fj(;)14 b Fs(1\))32 b Fh(\000)-15 b(!)33 b Fj(H)2931 2591 y Fi(m)3027 2579 y Fs(b)r(e)g(the)h Fj(L)3351 2591 y Fr(2)523 2678 y Fs(orthogonal)25 b(pro)5 b(jection)27 b(on)n(to)g Fj(H)1589 2690 y Fi(m)1652 2678 y Fs(.)37 b(If)28 b(w)n(e)f(write)g(the)h(solution)g(of)f(\(7.1\))g(in) h(the)g(form)1773 2829 y Fj(u)22 b Fs(=)h Fj(p)18 b Fs(+)g Fj(q)s(;)523 2980 y Fs(where)27 b Fj(p)c Fs(=)g Fj(P)969 2992 y Fi(m)1032 2980 y Fj(u)k Fs(and)h Fj(q)e Fs(=)c(\()p Fj(I)k Fh(\000)18 b Fj(P)1649 2992 y Fi(m)1712 2980 y Fs(\))p Fj(u)p Fs(,)28 b(then)g(equation)f(\(7.1\))g(b)r(ecomes)994 3191 y Fj(p)1036 3203 y Fi(t)1148 3191 y Fs(=)82 b Fj(a)1339 3203 y Fr(0)1376 3191 y Fj(p)19 b Fs(+)f(\()p Fj(a)1596 3203 y Fr(1)1652 3191 y Fs(+)g Fj(ib)1800 3203 y Fr(1)1836 3191 y Fs(\))p Fj(p)1910 3203 y Fi(xx)2008 3191 y Fs(+)g(\()p Fj(a)2167 3203 y Fr(3)2223 3191 y Fs(+)g Fj(ib)2371 3203 y Fr(3)2407 3191 y Fs(\)\()2471 3078 y Fg(Z)2555 3098 y Fr(1)2518 3267 y(0)2606 3191 y Fh(j)p Fj(u)p Fh(j)2700 3157 y Fr(2)2737 3191 y Fj(dx)p Fs(\))p Fj(p)316 b Fs(\(7.2\))998 3430 y Fj(q)1035 3442 y Fi(t)1148 3430 y Fs(=)82 b Fj(a)1339 3442 y Fr(0)1376 3430 y Fj(q)22 b Fs(+)c(\()p Fj(a)1594 3442 y Fr(1)1650 3430 y Fs(+)g Fj(ib)1798 3442 y Fr(1)1834 3430 y Fs(\))p Fj(q)1903 3442 y Fi(xx)2002 3430 y Fs(+)g(\()p Fj(a)2161 3442 y Fr(3)2216 3430 y Fs(+)h Fj(ib)2365 3442 y Fr(3)2401 3430 y Fs(\)\()2465 3317 y Fg(Z)2549 3337 y Fr(1)2512 3505 y(0)2600 3430 y Fh(j)p Fj(u)p Fh(j)2694 3395 y Fr(2)2731 3430 y Fj(dx)p Fs(\))p Fj(q)s(:)301 b Fs(\(7.3\))523 3633 y(The)28 b(real)e(part)h(of)h(the)g Fj(L)1330 3603 y Fr(2)1394 3633 y Fs(inner)g(pro)r(duct)f(of)h(the)g (equation)f(\(7)p Fj(:)p Fs(3\))g(with)35 b(\026)-49 b Fj(q)31 b Fs(giv)n(es)1137 3771 y(1)p 1137 3808 V 1137 3884 a(2)1213 3771 y Fj(d)p 1198 3808 74 4 v 1198 3884 a(dt)1282 3827 y Fh(k)p Fj(q)s Fh(k)1406 3793 y Fr(2)1465 3827 y Fs(=)23 b Fj(a)1597 3839 y Fr(0)1634 3827 y Fh(k)p Fj(q)s Fh(k)1758 3793 y Fr(2)1812 3827 y Fh(\000)18 b Fj(a)1939 3839 y Fr(1)1977 3827 y Fh(k)p Fj(q)2056 3839 y Fi(x)2097 3827 y Fh(k)2139 3793 y Fr(2)2194 3827 y Fh(\000)g(j)p Fj(a)2344 3839 y Fr(3)2381 3827 y Fh(j)28 b(k)p Fj(u)p Fh(k)2564 3793 y Fr(2)2600 3827 y Fh(k)p Fj(q)s Fh(k)2724 3793 y Fr(2)2760 3827 y Fj(:)523 4011 y Fs(Since)g Fh(k)p Fj(q)819 4023 y Fi(x)860 4011 y Fh(k)902 3981 y Fr(2)962 4011 y Fh(\025)23 b Fs(\(2)p Fj(\031)s(m)p Fs(\))1279 3981 y Fr(2)1316 4011 y Fh(k)p Fj(q)s Fh(k)1440 3981 y Fr(2)1476 4011 y Fs(,)28 b(w)n(e)f(ha)n(v)n(e)1363 4150 y(1)p 1363 4187 42 4 v 1363 4263 a(2)1440 4150 y Fj(d)p 1425 4187 74 4 v 1425 4263 a(dt)1508 4206 y Fh(k)p Fj(q)s Fh(k)1632 4171 y Fr(2)1691 4206 y Fh(\024)c Fs(\()p Fj(a)1855 4218 y Fr(0)1911 4206 y Fh(\000)18 b Fj(a)2038 4218 y Fr(1)2075 4206 y Fs(\(2)p Fj(\031)s(m)p Fs(\))2304 4171 y Fr(2)2341 4206 y Fs(\))p Fh(k)p Fj(q)s Fh(k)2497 4171 y Fr(2)2534 4206 y Fj(:)660 b Fs(\(7.4\))523 4415 y(So)29 b(b)n(y)g(c)n(ho)r(osing)f Fj(m)d(>)1315 4383 y Fr(1)p 1294 4397 75 4 v 1294 4444 a(2)p Fi(\031)1378 4329 y Fg(q)p 1461 4329 89 4 v 1471 4381 a Fi(a)1507 4389 y Fe(0)p 1471 4396 69 4 v 1471 4444 a Fi(a)1507 4452 y Fe(1)1579 4415 y Fs(w)n(e)k(get)g Fh(k)p Fj(q)s Fh(k)24 b(!)i Fs(0)j(exp)r(onen)n(tially)-7 b(.)41 b(Moreo)n(v)n(er,)27 b(supp)r(ose)523 4546 y Fj(q)560 4558 y Fr(0)620 4546 y Fs(=)c(0,)j Fj(p)841 4558 y Fr(0)901 4546 y Fh(2)e Fj(H)1049 4558 y Fi(m)1138 4546 y Fs(then)j(if)g Fj(u)22 b Fs(=)h Fj(p)16 b Fs(+)f Fj(q)30 b Fs(is)c(the)g(solution)g(of)h(\(7)p Fj(:)p Fs(1\))f(with)g Fj(u)2826 4558 y Fr(0)2886 4546 y Fs(=)d Fj(p)3016 4558 y Fr(0)3053 4546 y Fs(,)k(w)n(e)f(ha)n(v)n(e) 523 4645 y Fj(q)s Fs(\()p Fj(t)p Fs(\))e Fh(\021)e Fs(0)28 b(with)g Fj(u)p Fs(\()p Fj(t)p Fs(\))23 b(=)f Fj(p)p Fs(\()p Fj(t)p Fs(\))28 b(where)f Fj(p)p Fs(\()p Fj(t)p Fs(\))h(solv)n(es)1064 4856 y Fj(p)1106 4868 y Fi(t)1158 4856 y Fs(=)23 b Fj(a)1290 4868 y Fr(0)1327 4856 y Fj(p)18 b Fs(+)g(\()p Fj(a)1546 4868 y Fr(1)1602 4856 y Fs(+)g Fj(ib)1750 4868 y Fr(1)1786 4856 y Fs(\))p Fj(p)1860 4868 y Fi(xx)1958 4856 y Fs(+)g(\()p Fj(a)2117 4868 y Fr(3)2173 4856 y Fs(+)g Fj(ib)2321 4868 y Fr(3)2358 4856 y Fs(\)\()2422 4743 y Fg(Z)2505 4763 y Fr(1)2468 4932 y(0)2556 4856 y Fh(j)p Fj(p)p Fh(j)2644 4822 y Fr(2)2682 4856 y Fj(dx)p Fs(\))p Fj(p)1914 5147 y Fs(20)p eop %%Page: 21 21 21 20 bop 1775 614 a Fj(p)p Fs(\(0\))23 b(=)f Fj(p)2075 626 y Fr(0)2112 614 y Fj(:)523 760 y Fs(Th)n(us)28 b(the)h(linear)f (space)f Fj(H)1401 772 y Fi(m)1493 760 y Fs(is)h(in)n(v)-5 b(arian)n(t)27 b(under)i(the)g(solution.)38 b(Consequen)n(tly)28 b Fj(H)3240 772 y Fi(m)3331 760 y Fs(is)523 878 y(an)j(Inertial)g (Manifold)h(of)f(dimension)h(2)p Fj(m)20 b Fs(+)h(1)31 b(for)g(an)n(y)g Fj(m)e(>)2602 845 y Fr(1)p 2582 859 75 4 v 2582 906 a(2)p Fi(\031)2666 791 y Fg(q)p 2749 791 89 4 v 2759 844 a Fi(a)2795 852 y Fe(0)p 2759 859 69 4 v 2759 906 a Fi(a)2795 914 y Fe(1)2837 878 y Fs(.)49 b(On)31 b(the)h(other)523 1003 y(hand,)23 b(to)e(\014nd)i(a)e(lo)n(w)n (er)f(b)r(ound)i(on)f(the)h(dimension)g(of)f(the)h(global)f(attractor)f (w)n(e)h(linearize)523 1103 y(\(7)p Fj(:)p Fs(1\))27 b(around)g(the)h(steady)f(state)g(solution)g Fj(u)1972 1115 y Fi(s)2031 1103 y Fh(\021)22 b Fs(0)27 b(to)h(obtain)f(the)h (linear)f(equation)1514 1280 y Fj(v)1554 1292 y Fi(t)1606 1280 y Fs(=)c Fj(a)1738 1292 y Fr(0)1775 1280 y Fj(v)f Fs(+)c(\()p Fj(a)1996 1292 y Fr(1)2051 1280 y Fs(+)g Fj(ib)2199 1292 y Fr(1)2236 1280 y Fs(\))p Fj(Av)s(:)821 b Fs(\(7.5\))523 1457 y(Notice)28 b(that)h Fj(v)s Fs(\()p Fj(t)p Fs(\))c(=)f Fj(A)1278 1469 y Fi(k)1320 1457 y Fs(\()p Fj(t)p Fs(\))p Fj(e)1453 1427 y Fr(2)p Fi(\031)r(k)q(xi)1657 1457 y Fs(with)29 b Fj(A)1909 1469 y Fi(k)1950 1457 y Fs(\(0\))c Fh(6)p Fs(=)f(0)k(is)g(an)g(unstable)h(solution)f(for)g(\(7) p Fj(:)p Fs(5\))523 1579 y(for)23 b(ev)n(ery)f Fh(j)p Fj(k)s Fh(j)h Fj(<)1094 1546 y Fr(1)p 1073 1560 75 4 v 1073 1607 a(2)p Fi(\031)1157 1493 y Fg(q)p 1240 1493 89 4 v 1250 1545 a Fi(a)1286 1553 y Fe(0)p 1250 1560 69 4 v 1250 1607 a Fi(a)1286 1615 y Fe(1)1329 1579 y Fs(.)35 b(Hence)24 b(follo)n(wing)e(Babin)h(&)g(Vishik)g([1992])f (\(see)h(also)f(T)-7 b(emam)523 1709 y([1988]\),)42 b(the)e(dimension)g (of)g(the)h(unstable)f(manifold)g(around)f Fj(u)2700 1721 y Fi(s)2779 1709 y Fh(\021)k Fs(0)d(is)g(larger)e(or)523 1831 y(equal)29 b(to)h(2)p Fj(m)963 1843 y Ff(\003)1020 1831 y Fs(+)20 b(1)29 b(where)g Fj(m)1491 1843 y Ff(\003)1556 1831 y Fs(=)1678 1798 y Fr(1)p 1657 1812 75 4 v 1657 1860 a(2)p Fi(\031)1741 1745 y Fg(q)p 1824 1745 89 4 v 1834 1797 a Fi(a)1870 1805 y Fe(0)p 1834 1812 69 4 v 1834 1860 a Fi(a)1870 1868 y Fe(1)1913 1831 y Fs(,)h(whic)n(h)g(also) f(forms)g(a)g(lo)n(w)n(er)f(b)r(ound)j(on)e(the)523 1957 y(dimension)22 b(of)f(the)h(global)f(attractor)f(and)h(of)h(the)g (Inertial)f(Manifold.)35 b(>F)-7 b(rom)22 b(the)g(ab)r(o)n(v)n(e,)523 2057 y(w)n(e)27 b(conclude)h(the)g(follo)n(wing)e(theorem:)523 2233 y Fo(Theorem)k(4)42 b Fn(If)32 b Fj(a)1151 2245 y Fr(0)1216 2233 y Fj(>)27 b Fs(0)p Fn(,)32 b Fj(a)1451 2245 y Fr(1)1516 2233 y Fj(>)27 b Fs(0)p Fn(,)32 b Fj(a)1751 2245 y Fr(2)1816 2233 y Fs(=)26 b Fj(b)1943 2245 y Fr(2)2008 2233 y Fs(=)h(0)k Fn(and)i Fj(a)2381 2245 y Fr(3)2445 2233 y Fj(<)27 b Fs(0)p Fn(,)33 b(then)f(the)g(line)l(ar)h(sp)l(ac)l(e) 523 2351 y Fj(H)592 2363 y Fi(m)651 2371 y Fc(\003)690 2351 y Fn(,)d(wher)l(e)g Fj(m)1052 2363 y Ff(\003)1114 2351 y Fs(=)22 b(2[)1296 2318 y Fr(1)p 1276 2332 75 4 v 1276 2379 a(2)p Fi(\031)1360 2265 y Fg(q)p 1443 2265 89 4 v 1453 2317 a Fi(a)1489 2325 y Fe(0)p 1453 2332 69 4 v 1453 2379 a Fi(a)1489 2387 y Fe(1)1531 2351 y Fs(])d(+)f(1)p Fn(,)29 b(is)h(the)g(Inertial)h(Manifold)h(of)e(lowest)h (dimension.)523 2624 y Fk(8)135 b(Conclusion)45 b(and)g(Discussion)523 2806 y Fs(In)f(this)g(pap)r(er,)k(w)n(e)c(ha)n(v)n(e)e(discussed)i(the) g(e\013ect)h(of)e(nonlo)r(cal)h(in)n(teractions)e(on)i(the)523 2906 y(Ginzburg-Landau)24 b(dynamics)i(b)n(y)f(studying)h(a)f(nonlo)r (cal)h(Ginzburg-Landau)e(t)n(yp)r(e)i(am-)523 3005 y(plitude)39 b(equation.)69 b(W)-7 b(e)39 b(can)f(see)g(that,)j(dep)r(ending)e(on)f (the)h(v)-5 b(alue)38 b(and)h(the)f(sign)g(of)523 3105 y Fj(a)567 3117 y Fr(3)604 3105 y Fs(,)i(the)e(nonlo)r(cal)e(term,)k (has)d(the)h(stabilizing/destabilizing)e(e\013ect.)67 b(In)37 b(fact,)j(when)523 3205 y Fj(a)567 3217 y Fr(3)629 3205 y Fj(>)25 b Fh(\000)p Fj(a)828 3217 y Fr(2)890 3205 y Fj(>)g Fs(0)k(the)g(nonlo)r(cal)f(term)h(destabilizes)f(the)i(system) e(to)r(o)h(m)n(uc)n(h)g(that)g(certain)523 3304 y(solutions)k(blo)n(w)g (up)g(in)h(\014nite)g(time)g(\(see)f(\(2.3\)\).)55 b(Ho)n(w)n(ev)n(er) 31 b(if)j Fj(a)2638 3316 y Fr(2)2698 3304 y Fs(+)22 b Fj(a)2829 3316 y Fr(3)2899 3304 y Fj(<)32 b Fs(0,)i(w)n(e)g(ha)n(v)n(e) 523 3404 y(global)d(existence)i(along)e(with)i(man)n(y)f(other)g (\014ner)g(features)g(suc)n(h)g(as)g(the)h(existence)f(of)523 3504 y(global)24 b(attractor)g(in)h Fj(L)1265 3473 y Fr(2)1302 3504 y Fs(\(0)p Fj(;)14 b Fs(1\))25 b(and)g(the)g(p)r (ossession)f(of)h(inertial)g(manifolds.)36 b(W)-7 b(e)25 b(ha)n(v)n(e)f(in)523 3603 y(this)32 b(case)e(obtained)h(the)g(upp)r (er)h(and)f(lo)n(w)n(er)e(b)r(ound)j(for)e(the)i(dimension)f(estimates) g(for)523 3703 y(the)25 b(global)f(attractor)f(and)i(also)e(the)i (dimension)g(of)g(inertial)f(manifold)h(for)f(the)h(nonlo)r(cal)523 3802 y(amplitude)35 b(equation)f(\(1.1\).)59 b(W)-7 b(e)35 b(further)f(compared)g(the)h(estimates)g(that)g(w)n(e)f(found)523 3902 y(with)27 b(that)g(of)f(the)h(corresp)r(onding)e(cubic)i (Ginzburg-Landau)e(equation)h(\(1.4\).)36 b(W)-7 b(e)27 b(\014nd)523 4002 y(that)33 b(if)h Fj(a)834 4014 y Fr(3)903 4002 y Fj(<)e Fs(0,)i(then)g(the)f(dimension)g(estimates)g(that)g(w)n (e)g(found)g(are)f(smaller)g(when)523 4101 y(compared)23 b(to)i(the)g(cubic)g(Ginzburg-Landau)e(equation;)i(whic)n(h)f (indicates)h(a)f(stabilizing)523 4201 y(e\013ect)g(of)g(the)g(nonlo)r (cal)f(term)h(in)g(this)g(case.)34 b(On)24 b(the)g(other)f(hand,)i (when)e Fj(a)2917 4213 y Fr(3)2978 4201 y Fj(>)f Fs(0)i(w)n(e)f(\014nd) 523 4301 y(that)j(the)h(dimension)f(estimates)f(for)h(the)g(nonlo)r (cal)f(equation)h(is)f(larger;)g(whic)n(h)h(signi\014es)523 4400 y(destabilization)f(due)g(to)h(the)g(nonlo)r(cal)e(term.)36 b(It)26 b(is)f(w)n(orth)g(men)n(tioning)g(that)h(the)g(ab)r(o)n(v)n(e) 523 4500 y(conclusion)e(is)g(based)h(on)f(global)g(energy)f(estimate)i (around)f(zero.)34 b(F)-7 b(or)25 b(instance,)g(w)n(e)f(did)523 4599 y(not)k(study)g(here)g(the)g(e\013ect)h(of)f Fj(b)1575 4611 y Fr(3)1640 4599 y Fs(on)f(the)i(nature)e(of)h(the)h(dynamics)e (while)i(sa)n(y)d Fj(a)3173 4611 y Fr(3)3234 4599 y Fs(=)e(0.)523 4699 y(The)g(sp)r(ecial)g(case)g Fj(a)1174 4711 y Fr(2)1234 4699 y Fs(=)f Fj(b)1358 4711 y Fr(2)1417 4699 y Fs(=)g(0)h(has)g(b)r (een)g(treated)g(in)h(section)f(7)f(and)i(an)f(explicit)g(linear)523 4799 y(Inertial)30 b(Manifold,)i(whic)n(h)f(has)f(a)h(dimension)f (equal)h(to)f(the)i(dimension)e(of)h(the)h(global)523 4898 y(attractor)26 b(w)n(as)h(found,)h(hence)f(it)h(is)g(of)f(lo)n(w)n (est)g(dimension.)1914 5147 y(21)p eop %%Page: 22 22 22 21 bop 648 614 a Fo(Ac)m(kno)m(wledgemen)m(t.)648 714 y Fs(W)-7 b(e)30 b(w)n(ould)f(lik)n(e)h(to)g(thank)f(the)i(anon)n (ymous)d(referees)h(for)g(their)h(helpful)h(commen)n(ts)523 814 y(and)38 b(suggestions.)67 b(This)38 b(w)n(ork)f(w)n(as)g (partially)h(supp)r(orted)g(b)n(y)g(the)g(ONR)h(gran)n(t)d(No.)523 913 y(N00014-89-J-3023,)f(the)k(NSF)g(gran)n(t)f(No.)70 b(DMS-9308774,)38 b(and)h(the)g(Univ)n(ersit)n(y)e(of)523 1013 y(California{Los)19 b(Alamos)i(National)g(Lab)r(oratory)e(INCOR)i (program.)33 b(P)n(art)20 b(of)h(this)h(w)n(ork)523 1112 y(w)n(as)e(done)g(when)h(E.S.)f(Titi)h(w)n(as)f(visiting)g(the)h (Mettag-Le\017er)e(Institute)j(in)f(Sw)n(eden,)h(and)523 1212 y(the)35 b(Departmen)n(t)g(of)g(Mathematics)f(at)h(the)g(T)-7 b(ec)n(hnion)34 b(in)h(Israel.)57 b(E.S.)34 b(Titi)i(also)d(ac-)523 1312 y(kno)n(wledges)25 b(the)i(supp)r(ort)f(of)h(the)f(Univ)n(ersit)n (y)g(of)g(California{Irvine)f(F)-7 b(acult)n(y)26 b(Researc)n(h)523 1411 y(F)-7 b(ello)n(wship.)523 1683 y Fk(References)565 1865 y Fs([1])41 b(A.)29 b(V.)g(Babin)f(and)h(M.)g(I.)f(Vishik,)i(\\)p Fn(A)n(ttr)l(actors)f(of)i(evolution)h(e)l(quations")p Fs(,)d(North-)694 1965 y(Holland,)e(Amsterdam,)h(1992.)565 2125 y([2])41 b(M.)33 b(Bartuccelli,)h(P)-7 b(.)33 b(Constan)n(tin,)h (C.)f(R.)h(Do)r(ering,)g(J.)f(D.)h(Gibb)r(on)f(and)g(M.)h(Gis-)694 2224 y(selfalt,)51 b(\\)p Fn(On)46 b(the)i(p)l(ossibility)h(of)f(soft)g (and)f(har)l(d)i(turbulenc)l(e)d(in)h(the)h(c)l(omplex)694 2324 y(Ginzbur)l(g-L)l(andau)29 b(e)l(quation)-8 b(")p Fs(,)28 b(Ph)n(ysica)e(D)i Fo(44)f Fs(\(1990\),)f(421-444.)565 2484 y([3])41 b(A.)22 b(M.)f(Blo)r(c)n(h)g(and)h(E.S.)f(Titi,)i(\\)p Fn(On)g(the)h(dynamics)i(of)f(the)f(r)l(otating)g(elastic)h(b)l(e)l (ams")p Fs(,)694 2583 y(Pro)r(ceedings)39 b(of)i(the)g(Conference)f (on:)64 b(New)41 b(T)-7 b(rends)40 b(in)i(Systems)f(Theory)-7 b(,)43 b(A.)694 2683 y(P)n(erdon)26 b(and)h(B.)h(F.)g(Wyman)f(\(eds\),) h(Birkhauser,)e(Boston)h(\(July)h(9-11,)d(1990\).)565 2843 y([4])41 b(B.)25 b(Co)r(c)n(kburn,)g(D.)h(Jones)e(and)h(E.S.)h (Titi,)g(\\)p Fn(De)l(gr)n(\023)-40 b(es)27 b(de)h(lib)l(ert)n(\023)-40 b(e)28 b(d)n(\023)-40 b(eterminants)28 b(p)l(our)692 2943 y(\023)-40 b(equations)34 b(nonlin)n(\023)-40 b(eair)l(es)35 b(dissip)l(atives",)i(\\)p Fs(Determining)32 b(degrees)e(of)i(freedom)g (for)694 3042 y(nonlinear)22 b(dissipativ)n(e)g(equations",)h(Comptes)g (Rendus)g(De)g(L'Acad)n(\023)-39 b(emie)22 b(Des)i(Sci-)694 3142 y(ences,)j(P)n(aris,)f(S)n(\023)-39 b(erie)26 b(I,)i Fo(321)p Fs(,)f(\(1995\),)f(\(to)i(app)r(ear\).)565 3302 y([5])41 b(B.)18 b(Co)r(c)n(kburn,)i(D.)f(Jones)f(and)g(E.S.)h(Titi,)h (\\Estimating)e(the)h(n)n(um)n(b)r(er)f(of)h(asymptotic)694 3401 y(degrees)26 b(of)i(freedom)f(for)g(nonlinear)f(dissipativ)n(e)h (systems",)g(Preprin)n(t,)f(1995.)565 3561 y([6])41 b(P)-7 b(.)28 b(Collet)g(\\)p Fn(Thermo)l(dynamic)33 b(limit)e(of)g(the)g (Ginzbur)l(g-L)l(andau)f(e)l(quations")p Fs(,)f(Non-)694 3661 y(linearit)n(y)d Fo(7)i Fs(\(1994\),)e(1175-1190.)565 3821 y([7])41 b(P)-7 b(.)43 b(Constan)n(tin,)k(C.)d(F)-7 b(oias,)47 b(\\)p Fn(Navier-Stokes)d(Equations")p Fs(,)k(The)c(Univ)n (ersit)n(y)e(of)694 3920 y(Chicago)26 b(Press,)g(1988.)565 4080 y([8])41 b(P)-7 b(.)24 b(Constan)n(tin,)h(C.)f(F)-7 b(oias,)25 b(B.)f(Nicolaenk)n(o)f(and)i(R.)f(T)-7 b(emam,)26 b(\\)p Fn(Inte)l(gr)l(al)g(Manifolds)694 4180 y(and)c(Inertial)h (Manifolds)i(for)e(Dissip)l(ative)h(Partial)f(Di\013er)l(ential)g (Equations")p Fs(,)f(AMS)694 4280 y Fo(70)27 b Fs(Springer-V)-7 b(erlag,)25 b(1989.)565 4440 y([9])41 b(P)-7 b(.)38 b(Constan)n(tin,)j (C.)e(F)-7 b(oias,)41 b(B.)d(Nicolaenk)n(o)f(and)i(R.)g(T)-7 b(emam,)41 b(\\)p Fn(Sp)l(e)l(ctr)l(al)f(b)l(arri-)694 4539 y(ers)31 b(and)h(inertial)h(manifolds)g(for)g(dissip)l(ative)h(p)l (artial)f(di\013er)l(ential)f(e)l(quations")p Fs(,)f(J.)694 4639 y(Dynamics)c(and)h(Di\013.)g(Eqns.)f Fo(1)h Fs(\(1989\),)e(45-73.) 523 4799 y([10])41 b(P)-7 b(.)30 b(Constan)n(tin,)i(C.)f(F)-7 b(oias)30 b(and)h(R.)g(T)-7 b(emam,)32 b(\\)p Fn(A)n(ttr)l(actor)g(r)l (epr)l(esenting)h(turbulent)694 4898 y(\015ows")p Fs(,)27 b(Memoirs)g(of)h(AMS)g Fo(53)f Fs(No.)h(314)e(\(1985\).)1914 5147 y(22)p eop %%Page: 23 23 23 22 bop 523 614 a Fs([11])41 b(A.)32 b(Debussc)n(he)f(and)h(R.)g(T)-7 b(emam,)33 b(\\)p Fn(Inertial)h(manifolds)h(and)f(their)g(dimensions")p Fs(,)694 714 y(in)26 b(\\Dynamical)g(Systems:)36 b(Theory)26 b(and)g(Applications",)g(S.)h(I.)g(Andersson,)f(A.)h(E.)694 814 y(A)h(ndersosn)e(and)i(U.)g(Ottoson)e(\(eds.\),)i(W)-7 b(orld)28 b(Scien)n(ti\014c,)g(1993.)523 980 y([12])41 b(A.)29 b(Do)r(elman)h(and)f(E.S.)g(Titi,)h(\\)p Fn(On)g(the)i(exp)l (onential)g(r)l(ate)f(of)h(c)l(onver)l(genc)l(e)g(of)g(the)694 1079 y(Galerkin)44 b(appr)l(oximation)i(in)d(the)g(Ginzbur)l(g{L)l (andau)i(e)l(quation)-8 b(")p Fs(,)45 b(Pro)r(ceedings)694 1179 y(of)33 b(the)g(NA)-7 b(TO)34 b(Adv)-5 b(anced)33 b(Researc)n(h)f(W)-7 b(orkshop:)46 b(Asymptotic)34 b(and)f(Numerical) 694 1279 y(Metho)r(ds)k(for)f(P)n(artial)f(Di\013eren)n(tial)i (Equations)e(with)i(Critical)g(P)n(arameters,)f(M.)694 1378 y(Garb)r(ey)c(and)i(H.G.)g(Kap)r(er)e(\(eds.\),)j(Klu)n(w)n(er)d (Academic)i(Publishers,)g(Dordrec)n(h)n(t,)694 1478 y(1993,)25 b(241-252.)523 1644 y([13])41 b(A.)28 b(Do)r(elman)f(and)g(E.S.)g (Titi,)h(\\)p Fn(R)l(e)l(gularity)h(of)i(solutions)f(and)g(the)f(c)l (onver)l(genc)l(e)h(of)694 1743 y(the)k(Galerkin)i(metho)l(d)f(in)f (the)h(Ginzbur)l(g{L)l(andau)g(e)l(quation)-8 b(")p Fs(,)33 b(Numerical)f(F)-7 b(unc-)694 1843 y(tional)27 b(Analysis)g(and)g (Optimization)h Fo(14)f Fs(\(1993\),)f(299-321.)523 2009 y([14])41 b(C.)25 b(R.)g(Do)r(ering,)g(J.)g(D.)h(Gibb)r(on,)g(D.)g(D.)f (Holm)h(and)e(B.)i(Nicolaenk)n(o,)e(\\)p Fn(L)l(ow)j(dimen-)694 2109 y(sional)32 b(b)l(ehavior)h(in)e(the)g(c)l(omplex)h(Ginzbur)l(g-L) l(andau)f(e)l(quation)-8 b(")p Fs(,)29 b(Nonlinearit)n(y)f Fo(1)694 2208 y Fs(\(1988\),)e(279-309.)523 2374 y([15])41 b(C.)30 b(R.)g(Do)r(ering,)h(J.)f(D.)g(Gibb)r(on)h(and)f(C.)g(Da)n(vid) g(Lev)n(ermore,)f(\\)p Fn(We)l(ak)j(and)h(str)l(ong)694 2474 y(solutions)28 b(of)i(the)e(c)l(omplex)h(Ginzbur)l(g-L)l(andau)g (e)l(quation)-8 b(")p Fs(,)26 b(Ph)n(ysica)e(D)j Fo(71)f Fs(\(1994\),)694 2574 y(285-318.)523 2740 y([16])41 b(F.)33 b(J.)f(Elmer,)h(\\)p Fn(Nonline)l(ar)i(and)g(nonlo)l(c)l(al)g(dynamics) g(of)h(sp)l(atial)t(ly)g(extende)l(d)e(sys-)694 2839 y(tems:)48 b(stationary)35 b(states,)h(bifur)l(c)l(ations)f(and)g (stability")p Fs(,)h(Ph)n(ysica)31 b(D)i Fo(30)g Fs(\(1988\),)694 2939 y(321-341.)523 3105 y([17])41 b(A.)33 b(B.)f(F)-7 b(errari)31 b(and)i(E.)f(S.)h(Titi,)i(\\)p Fn(Gevr)l(ey)f(r)l(e)l (gularity)h(of)h(solutions)e(of)h(a)g(class)g(of)694 3205 y(analytic)c(nonline)l(ar)f(p)l(ar)l(ab)l(olic)i(e)l(quations")p Fs(,)27 b(Preprin)n(t,)g(1994.)523 3371 y([18])41 b(C.)31 b(F)-7 b(oias,)32 b(G.)g(Sell)g(and)f(R.)h(T)-7 b(emam,)33 b(\\)p Fn(Inertial)g(manifolds)j(for)e(nonline)l(ar)g(evolu-)694 3470 y(tionary)c(e)l(quations")p Fs(,)e(Jour.)f(Di\013.)h(Eqns.)f Fo(73)g Fs(\(1988\),)g(309-353.)523 3636 y([19])41 b(C.)30 b(F)-7 b(oias,)30 b(G.)g(Sell)g(and)g(E.)g(Titi,)h(\\)p Fn(Exp)l(onential)h(tr)l(acking)h(and)f(appr)l(oximation)i(of)694 3736 y(inertial)j(manifolds)h(for)f(dissip)l(ative)i(nonline)l(ar)e(e)l (quations")p Fs(,)f(J.)f(Dynamics)f(Di\013.)694 3836 y(Eqns.)27 b Fo(1)g Fs(\(1989\),)f(199-244.)523 4002 y([20])41 b(C.)34 b(F)-7 b(oias)33 b(and)h(R.)g(T)-7 b(emam,)36 b(\\)p Fn(Some)f(analytic)i(and)f(ge)l(ometric)h(pr)l(op)l (erties)g(of)f(the)694 4101 y(solutions)27 b(of)i(the)e(Navier-Stokes)h (e)l(quations")p Fs(,)e(J.)f(Math.)g(Pures)f(Appl.)i Fo(58)f Fs(\(1979\),)694 4201 y(339-368.)523 4367 y([21])41 b(C.)27 b(F)-7 b(oias,)27 b(R.)h(T)-7 b(emam,)27 b(\\)p Fn(Determination)j(of)g(the)g(solutions)g(of)h(the)f(Navier-Stokes)694 4467 y(e)l(quations)g(by)g(a)g(set)f(of)i(no)l(dal)g(values")p Fs(,)d(Math.)f(Comput.)h Fo(43)f Fs(\(1984\),)g(117-133.)523 4633 y([22])41 b(C.)33 b(F)-7 b(oias)32 b(and)g(R.)h(T)-7 b(emam,)35 b(\\)p Fn(Gevr)l(ey)f(class)i(r)l(e)l(gularity)f(for)g(the)g (solutions)g(of)g(the)694 4732 y(Navier-Stokes)30 b(e)l(quations")p Fs(,)e(J.)f(F)-7 b(unctional)28 b(Anal.)g Fo(87)f Fs(\(1989\),)f (359-369.)1914 5147 y(23)p eop %%Page: 24 24 24 23 bop 523 614 a Fs([23])41 b(C.)30 b(F)-7 b(oias)28 b(and)i(E.S.)g(Titi,)g(\\)p Fn(Determining)i(no)l(des,)h(\014nite)e (di\013er)l(enc)l(e)h(schemes)h(and)694 714 y(inertial)d(manifolds")p Fs(,)g(Nonlinearit)n(y)c Fo(4)i Fs(\(1991\),)e(135-153.)523 880 y([24])41 b(J.)26 b(M.)h(Ghidaglia)e(and)h(B.)h(H)n(\023)-39 b(eron,)25 b(\\)p Fn(Dimensions)j(of)i(the)f(attr)l(actors)f(asso)l (ciate)l(d)i(to)694 980 y(the)g(Ginzbur)l(g-L)l(andau)f(e)l(quation)-8 b(")p Fs(,)28 b(Ph)n(ysica)e(D)i Fo(28)f Fs(\(1987\),)f(282-304.)523 1146 y([25])41 b(J.)20 b(K.)h(Hale,)h(\\)p Fn(Asymptotic)h(Behavior)j (of)e(Dissip)l(ative)h(Systems")p Fs(,)d(American)e(Math.)694 1245 y(So)r(c.,)27 b(\(1988\).)523 1411 y([26])41 b(D.A.)25 b(Jones)e(and)h(E.S.)h(Titi,)g(\\)p Fn(On)h(the)h(numb)l(er)e(of)j (determining)g(no)l(des)f(for)g(the)g(2D)694 1511 y(Navier-Stokes)j(e)l (quations")p Fs(,)e(J.)f(Math.)h(Anal.)g(Appl.)g Fo(168)f Fs(\(1992\),)f(72-88.)523 1677 y([27])41 b(D.A.)33 b(Jones)f(and)g (E.S.)g(Titi,)i(\\)p Fn(Determination)g(of)h(the)g(solutions)f(of)h (the)g(Navier-)694 1777 y(Stokes)25 b(e)l(quations)h(by)h(\014nite)e (volume)h(elements")p Fs(,)e(Ph)n(ysica)e(D)i Fo(60)f Fs(\(1992\),)f(165-174.)523 1943 y([28])41 b(D.A.)c(Jones)f(and)g(E.S.) h(Titi,)i(\\)p Fn(Upp)l(er)f(b)l(ounds)g(on)g(the)g(numb)l(er)g(of)h (determining)694 2042 y(mo)l(des,)28 b(no)l(des,)h(and)f(volume)f (elements)h(for)g(the)f(Navier-Stokes)h(e)l(quations")p Fs(,)d(Indi-)694 2142 y(ana)i(Math.)g(Jour.)g Fo(42)g Fs(\(1993\),)f(875-887.)523 2308 y([29])41 b(I.)20 b(Kuk)-5 b(a)n(vica,)21 b(\\)p Fn(On)h(the)h(numb)l(er)g(of)h(determining)g(no)l (des)g(for)g(the)f(Ginzbur)l(g-L)l(andau)694 2408 y(e)l(quation)-8 b(")p Fs(,)27 b(Nonlinearit)n(y)g Fo(5)g Fs(\(1992\),)f(997-1006.)523 2574 y([30])41 b(C.D.)35 b(Lev)n(ermore)d(and)i(M.)g(Oliv)n(er,)h(\\)p Fn(The)i(c)l(omplex)f(Ginzbur)l(g-L)l(andau)g(e)l(quation)694 2673 y(as)42 b(a)g(mo)l(del)h(pr)l(oblem")p Fs(,)i(Preprin)n(t)39 b(1995,)j(\(T)-7 b(o)41 b(app)r(ear)e(in)i(Lecture)g(in)g(Applied)694 2773 y(Mathematics,)27 b(AMS.\))523 2939 y([31])41 b(B.)29 b(J.)g(Matk)n(o)n(wsky)e(and)i(V.)h(V)-7 b(olp)r(ert,)29 b(\\)p Fn(Couple)l(d)k(Ginzbur)l(g-L)l(andau)e(e)l(quations)g(in)694 3039 y(gasless)f(c)l(ombustion)-8 b(")p Fs(,)27 b(Ph)n(ysica)f(D)i Fo(54)g Fs(\(1992\),)e(203-219.)523 3205 y([32])41 b(P)-7 b(.)23 b(Metzener)g(and)g(M.)h(R.)f(E.)g(Pro)r(ctor,)g(\\)p Fn(Nonlo)l(c)l(al)j(e\013e)l(cts)f(on)h(p)l(attern)g(sele)l(ction)g(in) 694 3304 y(dir)l(e)l(ctional)i(solidi\014c)l(ation)-8 b(")p Fs(,)27 b(In)n(ter.)e(Conf.)g(on)f(Nonlinear)g(Dynamics)h(and)f (P)n(attern)694 3404 y(F)-7 b(ormation)26 b(in)i(Natural)f(En)n (vironmen)n(t,)f(the)i(Netherlands,)g(1994.)523 3570 y([33])41 b(A.)26 b(Mielk)n(e,)g(\\)p Fn(R)l(e)l(ductions)g(of)j(PDEs)g (on)f(domains)h(with)f(sever)l(al)h(unb)l(ounde)l(d)f(dir)l(e)l(c-)694 3670 y(tions:)46 b(a)34 b(\014rst)e(step)i(towar)l(ds)g(mo)l (dulational)h(e)l(quations")p Fs(,)e(J.)e(Appl.)i(Math.)e(Ph)n(ys.)694 3769 y(\(ZAMP\))d Fo(43)f Fs(\(1992\),)f(449-470.)523 3935 y([34])41 b(A.)29 b(C.)h(New)n(ell,)g(T.)f(P)n(assot)f(and)h(J.)g (Lega,)g(\\)p Fn(Or)l(der)i(p)l(ar)l(ameter)h(e)l(quations)f(for)h(p)l (at-)694 4035 y(terns")p Fs(,)27 b(Ann)n(u.)h(Rev.)f(Fluid)h(Mec)n(h.)g Fo(25)f Fs(1993,)f(399-453.)523 4201 y([35])41 b(M.)29 b(Or-Guil,)g(M.)g(Bo)r(de)f(and)h(H.-G.)g(Purwins,)g(\\)p Fn(Nonlo)l(c)l(al)i(mo)l(dulation)h(e)l(quations)694 4301 y(describing)27 b(the)f(inter)l(action)h(of)g(Hopf)g(and)f(T)-6 b(uring)26 b(bifur)l(c)l(ations")p Fs(,)f(In)n(ter.)e(Conf.)h(on)694 4400 y(Nonlinear)29 b(Dynamics)h(and)g(P)n(attern)f(F)-7 b(ormation)29 b(in)i(Natural)e(En)n(vironmen)n(t,)h(the)694 4500 y(Netherlands,)d(1994.)523 4666 y([36])41 b(D.)20 b(T.)f(P)n(apageorgiou,)f(C.)h(Maldarelli)g(and)g(D.)h(S.)g(Rumsc)n (hitzki,)h(\\)p Fn(Nonline)l(ar)i(inter-)694 4765 y(facial)31 b(stability)g(of)g(c)l(or)l(e-annular)e(\014lm)h(\015ows")p Fs(,)e(Ph)n(ys.)e(Fluids)i Fo(2)f Fs(\(1990\),)f(340-352.)1914 5147 y(24)p eop %%Page: 25 25 25 24 bop 523 614 a Fs([37])41 b(A.)25 b(P)n(azy)-7 b(,)24 b(\\)p Fn(Semigr)l(oups)j(of)h(Line)l(ar)g(Op)l(er)l(ators)f(and)h (Applic)l(ations)g(to)f(Partial)i(Dif-)694 714 y(fer)l(ential)h (Equations")p Fs(,)e(Springer-V)-7 b(erlag,)26 b(Berlin,)h(1983.)523 880 y([38])41 b(G.)28 b(R.)h(Sell)f(and)h(Y.)f(C.)h(Y)-7 b(ou,)28 b(\\)p Fn(Inertial)j(manifolds|The)i(nonself-adjoint)f(c)l (ase")p Fs(,)694 980 y(J.)27 b(Di\013.)i(Eqn.)e Fo(96)g Fs(\(1992\),)f(203-255.)523 1146 y([39])41 b(P)-7 b(.)30 b(T)-7 b(ak\023)-42 b(a)n(\024)j(c,)29 b(P)-7 b(.)31 b(Bollerman,)f(A.)h(Do)r(elman,)g(A.)g(v)-5 b(an)31 b(Harten,)g(and)f (E.S.)h(Titi,)h(\\)p Fn(A)n(n-)694 1245 y(alyticity)42 b(of)f(essential)t(ly)h(b)l(ounde)l(d)e(solutions)h(to)f(str)l(ongly)h (p)l(ar)l(ab)l(olic)h(semiline)l(ar)694 1345 y(systems")p Fs(,)27 b(SIAM,)h(Jour.)f(Math.)g(Anal.)h(\(1995\))e(\(to)i(app)r (ear\).)523 1511 y([40])41 b(R.)c(T)-7 b(emam,)40 b(\\)p Fn(In\014nite)d(Dimensional)i(Dynamic)l(al)h(Systems)e(in)h(Me)l (chanics)h(and)694 1611 y(Physics")29 b Fs(,)e(Springer-V)-7 b(erlag,)26 b(Berlin,)h(1988.)523 1777 y([41])41 b(J.)23 b(M.)h(V)-7 b(ega,)24 b(\\)p Fn(On)h(the)h(amplitude)i(e)l(quations)e (arising)i(at)e(the)g(onset)g(of)h(the)f(oscil)t(la-)694 1876 y(tory)i(instability)h(in)f(p)l(attern)g(formation)-8 b(")p Fs(,)26 b(SIAM,)g(Jour.)f(Math.)h(Anal.)f Fo(24)h Fs(\(1993\),)694 1976 y(603-617.)523 2142 y([42])41 b(J.)20 b(W.)g(Wilder,)i(B.)e(F.)g(Edw)n(ards,)g(D.)h(A.)f(V)-7 b(asquez)20 b(and)f(G.)i(I.)f(Siv)-5 b(ashinsky)e(,)21 b(\\)p Fn(Deriva-)694 2242 y(tion)36 b(of)g(a)g(nonline)l(ar)g(fr)l (ont)g(evolution)g(e)l(quation)g(for)h(chemic)l(al)g(waves)g(involving) 694 2341 y(c)l(onve)l(ction)-8 b(")p Fs(,)27 b(Ph)n(ysica)f(D)i Fo(73)f Fs(\(1994\),)g(217-226.)1914 5147 y(25)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF