\documentstyle[12pt,amsfonts]{article}
\begin{document}
\title{Well-posedness of a Semilinear Heat \\Equation
with Weak Initial Data}
\author{Jiahong Wu
\\School of Mathematics\\The Institute for Advanced Study
\\Princeton, NJ 08540 }
\date{}
\maketitle
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{define}[thm]{Definition}
\newtheorem{rem}[thm]{Remark}
\newtheorem{example}[thm]{Example}
\newtheorem{lemma}[thm]{Lemma}
\def\theequation{\thesection.\arabic{equation}}
\begin{abstract}
This paper mainly consists of two parts. In the first part
the initial value problem (IVP)of the semilinear heat equation
$$
\partial_t u- \Delta u =|u|^{k-1}u,
\quad \mbox{on \quad ${\Bbb R}^n\times (0,\infty)$},\quad k\ge 2
$$
$$
u(x,0)=u_0(x),\qquad \qquad x\in{\Bbb R}^n\qquad\qquad\quad
$$
with initial data in $\dot{L}_{r,p}$ are studied. We prove the
well-posedness when
$$
1
0$) and
$$
\|u_\lambda(\cdot,t)\|_{\dot{L}_{r,p}}=\lambda^{r-\left(\frac{n}{p}-
\frac{2}{k-1}\right)}\|u(\cdot, \lambda^2 t)\|_{\dot{L}_{r,p}}
$$
But unfortunately, we don't know if dimensional analysis always
works and how to give a generally applicable criterion to detect the
indices.
\vspace{.13in}
The problem of well-posedness for semilinear heat equation has attracted
the attention of many mathematicians, but not many results related to
very weak initial data have been published.
The results concerning the IVP (\ref{eq:1.1}),(\ref{eq:1.2}) in $L^p$ setting obtained
by Weissler and others are as follows: for $p\ge \frac{n(k-1)}{2}$, there
is well-posedness (\cite{w1},\cite{w2});
for $1\le p<\frac{n(k-1)}{2}$, non-unique solutions
can be constructed \cite{hw}.
By letting $r=0$, i.e., $p=\frac{n(k-1)}{2}$, our results reduce to those
in the $L^p$ theory.
In \cite{bp} Baras and Pierre prove that the IVP
(\ref{eq:1.1}),(\ref{eq:1.2}) has a solution if and only if the initial
measure is not too much concentrated.
Clearly Sobolev spaces
of negative indices contain distributions.
In \cite{ky} Kozono and Yamazaki consider the IVP
(\ref{eq:1.1}),(\ref{eq:1.2}) with initial data in real interpolation
space ${\cal N}^{r}_{p,q,\infty}$ $(r=\frac{n}{p}-\frac{2}{k-1})$,
which is slightly larger than $\dot{L}_{r,p}$ (by noting that
${\cal N}^{r}_{p,p,\infty}
=\dot{B}^{r}_{p,\infty} \supseteq \dot{L}_{r,p}$,
where $\dot{B}^{r}_{p,\infty}$ is a homogeneous
Besov space, see e.g. \cite{bl} p.147). However they obtain
existence under the assumption that $k\le q\le p$ and
$n(k-1)<2p0)$ denotes the
class of all functions $u
\in C([0,T],H^s)\cap C((0,T], H^r)$ that also satisfy the condition
$$
\|u\|_{BC_s((0,T], H^r)}\equiv \sup_{t\in[0,T]}\|(1+|\xi|^2)^{\frac{s}{2}}
(1+|\xi|^2 t)^{\frac{r-s}{2}}\hat{u}(t)(\xi)\|_{L^2}< \infty
$$
Here $\hat{f}$ denotes the Fourier transform of $f$.
\vspace{.13in}
We prove that if $u_0\in H^s$ with $s$ satisfying
\begin{equation}\label{eqk:1.3}
-1< s,\quad \mbox{for $n=1$},\qquad \frac{n}{2}-20$ the
solution $u\in BC_s((0,T],H^r)$ for any $r\ge s$.
See Theorem \ref{thm:4.1} of Section \ref{sec:4} for a precise
statement. Clearly this result is not covered by the well-posedness
result in the first part.
\vspace{.13in}
As we explained before, the index $s=\frac{n}{2}-2$ for $n\ge 2$
is exactly the number from dimensional analysis. But for
$n=1$, our method only allow us to prove well-posedness for $s>-1$ and
fails to extend to $s>-3/2$. It would be desirable to show that
$s=-1$ is actually sharp by providing a counter example.
\vspace{.13in}
By taking special values of $s$ and $u_0$, the well-posedness in this
part reduces to some of those previously obtained by other authors
(\cite{bf},\cite{w1}). Letting $s=0$, our result (for $n\le 4$) reduces to
the $L^p$ theory of Weissler and others (\cite{w1},\cite{w2}). In \cite{bf}
Brezis and Friedman prove for $u_0=\delta(x)$ that
the solution exists for $0\frac{n}{2}-2$, which is slightly
more regular than $\delta(x)$ since
$\delta(x)\not \in H^{s}
({\Bbb R}^n) \quad\mbox{for} \quad s>\frac{n}{2}-2$.
The fact that for $n=2$,
$\delta(x)\not \in H^{-1}({\Bbb R}^2)$, but $\delta(x)\in H^{-1-\epsilon}({\Bbb R}^2)$
for any $\epsilon>0$, combined with the
their non-existence result implies that our well-posedness result in 2-D
is actually sharp.
\vspace{.12in}
The well-posedness result in this part is again
proved by the contraction mapping arguments and
we only deal with the IVP (\ref{eqk:1.1}),(\ref{eqk:1.2})
for $s\le 0$. The proof
for $s>0$
can be given in a similar (and actually easier) way.
\vspace{.2in}
It is a great pleasure to thank Professor Carlos Kenig for helpful
suggestions concerning this work. This work is supported by
NSF grant DMS 9304580 at IAS.
\newpage
\section{Well-posedness in $\dot{L}_{r,p}$}\label{sec:2}
\setcounter{equation}{0}
First we define the spaces of weighted continuous functions in time, which
have been introduced by Kato, Ponce and others in solving
the Navier-Stokes equations (\cite{k2},\cite{kp1},\cite{kp2}).
\begin{define}
Suppose $T>0$ and $\alpha\ge 0$ are real numbers. The spaces
$ C_{\alpha,s,q}$ and $\dot{C}_{\alpha,s,q}$ are defined as
$$
C_{\alpha,s,q} \equiv \{f \in C((0,T), \dot{L}_{s,q}), \quad
\|f\|_{\alpha,s,q} <\infty\}
$$
where the norm is given by
$$
\|f\|_{\alpha,s,q}=\sup \{t^\alpha \|f\|_{s,q}, \quad t\in (0,T)\}
$$
$\dot{C}_{\alpha,s,q}$ is a subspace of $C_{\alpha,s,q}$:
$$
\dot{C}_{\alpha,s,q}\equiv\{f\in C_{\alpha,s,q}, \quad \lim_{t\to 0}
t^\alpha\|f\|_{s,q}= 0\}
$$
When $\alpha=0$, $\bar{C}_{s,q}$ are used for $BC([0,T),\dot{L}_{s,q})$.
\end{define}
These spaces are important in uniqueness and local existence problems
(\cite{k2},\cite{kp1},\cite{kp2}).
$f\in C_{\alpha,s,q}$ (resp. $f\in \dot{C}_{\alpha,s,q}$) implies
that $\|f(t)\|_{s,q}=O(t^{-\alpha})$ (resp. $o(t^{-\alpha})$).
\vspace{.14in}
The main result of this section is the well-posedness theorem that states
\begin{thm}\label{thm:2.1}
Assume $u_0
\in \dot{L}_{r,p}$ with $p$ and $r$ satisfying
\begin{equation}\label{eq:p}
10$, there exists a unique
solution $u={\frak U}(u_0)$ to the IVP (\ref{eq:1.1}), (\ref{eq:1.2}) such that
\begin{equation}\label{eq:2.1}
u\in Y_T\equiv (\cap_{p\le q<\infty}\bar{C}_{r,q})\cap (\cap_{p\le q<\infty}
\cap_{s>r}\dot{C}_{(s-r)/2,s,q})
\end{equation}
In particular, (\ref{eq:2.1}) implies that
$$
u\in BC([0,T), \dot{L}_{r,p})\cap (\cap_{s>r} C((0,T), \dot{L}_{s,p})
$$
Furthermore, the mapping
$${\frak U}: \Lambda\longmapsto Y_T: \quad u_0\longmapsto u$$
is Lipschitz
in some neighborhood $\Lambda$ of $u_0$.
\end{thm}
We make several remarks concerning this theorem.
\vspace{.1in}
\begin{rem}
It is easy to see from our proof of this theorem
that if $\|u_0\|_{\dot{L}_{r,p}}$ is sufficiently small,
we may take $T=\infty$.
\end{rem}
\vspace{.1in}
\begin{rem}
The homogeneous spaces $\dot{L}_{s,q}$ can be replaced by
inhomogeneous
spaces $L_{s,q}$ (i.e., spaces of Bessel potentials):
$$
L_{s,q}\equiv \{ v: \|v\|_{s,q}\equiv\|(1-\Delta)^{s/2} u\|_{L^q} <\infty\}
$$
to obtain quite similar well-posedness results.
\end{rem}
\vspace{.2in}
We prove Theorem \ref{thm:2.1} by the method of integral equation and
contraction-mapping arguments.
This method has been extensively used by Kato, Ponce and others to prove the
well-posedness of the Navier-Stokes equations in various type of
functional spaces (\cite{k1},\cite{k2},\cite{kf},\cite{kp1},\cite{kp2}).
First we write Equation (\ref{eq:1.1}) in the integral form
$$
u=U u_0(t)+G(|u|^{k-1}u)(t)\equiv e^{-\Delta t}u_0 +\int_{0}^{t}
e^{-\Delta(t-\tau)}(|u|^{k-1}u)(\tau) d\tau
$$
Then we estimate the operators $U$ and $G$ separately. The main
estimates are established in the propositions that follow.
\begin{prop}\label{prop1}
\begin{description}
\item[(1)] If $s\in {\Bbb R}$ and $q\in [1,\infty)$, then
$$
U u_0(t)\to u_0, \quad \mbox{ in $\dot{L}_{s,q}$\quad as $t\to 0$}.
$$
\item[(2)] If $s_1\le s_2$, $q_1\le q_2$, and
$$
\alpha_2=\left(s_2-s_1 +\frac{n}{q_1}-\frac{n}{q_2}\right)
$$
then $U$ maps continuously from $\dot{L}_{s_1,q_1}$ into $\dot{C}_{
\alpha_2,s_2,q_2}$.
\end{description}
\end{prop}
{\bf Proof.}\quad The proof of (1) involves the definitions of the norms
and the dominated convergence theorem. See \cite{kp1},\cite{kf} for the proof of (2).
\vspace{.2in}
Now we give the estimates for the operator $G$:
$$
G g(t)=\int_{0}^{t}e^{-\Delta(t-\tau)}g(\tau) d\tau
$$
\begin{prop}\label{prop2}
If $q_1,q_2, \alpha_1,\alpha_2$, $s_1$ and $s_2$ satisfy
$$
q_1\le q_2,
$$
$$
\alpha_1<1,\quad \alpha_2=\alpha_1-1 +\frac{1}{2}\left[
s_2-s_1+\frac{n}{q_1}-\frac{n}{q_2}\right]
$$
$$
0\le s_2-s_1<2-{n}\left(\frac{1}{q_1}-\frac{1}{q_2}\right)
$$
then $G$ maps continuously from $\dot{C}_{\alpha_1,s_1,q_1}$ to
$\dot{C}_{\alpha_2,s_2,q_2}$.
\end{prop}
{\bf Proof.}\quad The proof of this proposition is quite similar to
that of Lemma 2.3 in \cite{kp1}. We just want to point out that the restrictions
$$
\alpha_1<1, \quad s_2-s_1<2-n\left(\frac{1}{q_1}-\frac{1}{q_2}\right)
$$
are imposed to guarantee the finiteness of a Beta function involved in
the estimates of $G$.
\vspace{.2in}
Now we turn to the proof of Theorem \ref{thm:2.1}.
\vspace{.1in}
\noindent{\bf Proof of Theorem \ref{thm:2.1}.}\quad
We consider two cases: $r<0$ and $r=0$. For $r<0$, we define
$$
X=\bar{C}_{r,p}\cap \dot{C}_{-\frac{r}{2},0,p},
$$
with norm for $u\in X$ given by
$$
\|u\|_X =\|u- U u_0\|_{0,r,p} +\|u\|_{-\frac{r}{2},0,p}
$$
and the complete metric space
$X_R$ to be the closed ball in $X$ of radius $R$.
Consider the operator ${\cal A}(u,u_0):X_R\times \Lambda\longmapsto X$
$$
{\cal A}(u,u_0)(t)= U u_0(t) + G (|u|^{k-1}u)(t),\quad 0\le t0$ is small enough and $\Lambda$ chosen properly.
To estimate $G$, we use Proposition \ref{prop2} with
$$
q_1=\frac{p}{k},\quad q_2=p,\quad \alpha_1=-\frac{k r}{2},\quad
\alpha_2=\frac{l}{k},\quad s_1=0,\quad s_2=\frac{2l}{k}+r,
$$
we obtain
$$
\|G (|u|^{k-1}u)\|_{\frac{l}{k},\frac{2l}{k}+r,p} \le c\||u|^{k-1}
u\|_{-\frac{kr}{2},
0,\frac{p}{k}}
\le \|u\|_{-\frac{r}{2},0,p}^{k} \le c R^k
$$
for all $l\in [0, -\frac{k^2}{2}r)$. Here it is important to
notice that the restrictions on $p$ and $r$ (\ref{eq:p}),
(\ref{eq:r}) are necessary
in order to apply Proposition \ref{prop2}.
\vspace{.12in}
Furthermore,
$$
\|{\cal A}(u,u_0)-{\cal A}({\tilde{u},u_0})\|_X \le
\|G (|u|^{k-1}u) - G (|\tilde{u}|^{k-1}\tilde{u})\|_X
$$
$$
\le \|G (|u-\tilde{ u}|^{k-1} u)\|_X + \|G (|u-\tilde{u}|
|\tilde{u}|^{k-1})\|_X
$$
Using Proposition \ref{prop2} again
$$
\|{\cal A}(u,u_0)-{\cal A}({\tilde{u},u_0})\|_X \le
2\||u-\tilde{u}|^{k-1}u\|_{-\frac{kr}{2}, 0, \frac{p}{k}}
+2\||u-\tilde{u}||\tilde{u}|^{k-1}\|_{-\frac{kr}{2}, 0, \frac{p}{k}}
$$
$$
\le c\|u\|_X\|u-\tilde{u}\|_{X}^{k-1} +c\|u-\tilde{u}\|_X \|\tilde{u}\|
_{X}^{k-1}
$$
So if we choose $T$ to be small and $R$ properly,
then ${\cal A}(u,u_0)$ maps $X_R$
into itself and is a contraction map when $k\ge2 $.
Consequently there exists a unique fixed point $u\in X_R$: $u={\frak U}
(u_0)$ satisfying $u={\cal A}(u,u_0)$.
It is easy to see from the above estimates that the uniqueness
can be extended to $X_{R'}$ for all $R'$ by reducing the time interval
and thus to the whole $X$.
\vspace{.1in}
To show that $u$ is in the class of $Y_T$, we notice
$$
u(t)={\cal A}(u,u_0)(t)\equiv U u_0(t) + G(|u|^{k-1} u)(t),\quad t\in [0,T).
$$
For small $s$, the above formula, combined with Proposition \ref{prop2}, can be used
to prove that $u\in Y_T$. For large $s$, $u\in Y_T$ can be shown by
induction (see an analogous argument in \cite{k2} (p.60)).
\vspace{.10in}
To prove the Lipschitz continuity of ${\frak U}$, let $u={\frak U}(u_0)$ and
$v={\frak U}(v_0)$ for $u_0, v_0\in \Lambda$. Then
$$
\|u-v\|_X =\|{\cal A}(u,u_0)-{\cal A}(v,v_0)\|_X
$$
$$
\le \|{\cal A}(u,u_0)-{\cal A}(v,u_0)\|_X +\|{\cal A}(v,u_0)-
{\cal A}(v,v_0)\|_X
$$
$$
\le \gamma\|u-v\|_X +\|U (u_0-v_0)\|_X
$$
For small $T$ and properly chosen $\Lambda$, $\gamma<1$ since the mapping
is a contraction and we obtain asserted result by using
Proposition \ref{prop1}
to the second term.
\vspace{.16in}
In the case $r=0$, we define
$$
X=\bar{C}_{0,p} \cap \dot{C}_{\frac{1}{4(k-1)}, 0,\frac{4p}{3}}
$$
with the norm
$$
\|u\|_X =\|u- U u_0\|_{0,0,p} + \|u\|_{\frac{1}{4(k-1)},0,\frac{4p}{3}}
$$
and $X_R$ is again the closed ball in $X$ of radius $R$. By Proposition
\ref{prop2},
$$
\| G (|u|^{k-1}u)\|_X = \| G(|u|^{k-1}u)\|_{0,0,p} +
\| G(|u|^{k-1}u)\|_{\frac{1}{4(k-1)},0,\frac{4p}{3}}
$$
$$
\le c \|u^k\|_{\frac{k}{4(k-1)},0,\frac{4p}{3k}}\le c\|u\|^{k}_{
\frac{1}{4(k-1)},0,\frac{4p}{3}} \le c R^k
$$
and the rest of the proof reduces to the previous case.
This completes the proof of Theorem \ref{thm:2.1}.
\newpage
\section{Non-uniqueness for $r<\frac{n}{p}-\frac{2}{k-1}$}\label{sec:3}
\setcounter{equation}{0}
In this section we consider the situation when
$$
1< p< \frac{n(k-1)}{2} 0$, there exists at least one non-trivial solution $\Phi$
to the IVP (\ref{eq:3.1}), (\ref{eq:3.2}) such that
$$
\Phi\in C([0,T), \dot{L}_{r,p})\cap C((0,T), \dot{C}_{-r/2,0,p})
$$
Thus we get at least three different solutions $\Phi$, $-\Phi$ and $0$,
corresponding to the same initial data $0$.
\end{thm}
\vspace{.2in}
We seek solutions to Equation (\ref{eq:3.1}) of the self-similar form
$$
\Phi(x,t)= t^{-\frac{1}{k-1}}\omega(\frac{x}{\sqrt{t}})
$$
Then the Equation (\ref{eq:3.1}) which $\Phi$ should satisfy reduces to an O.D.E. of
$\omega$,
$$
\Delta \omega(x) +\frac{x}{2}\cdot \nabla\omega(x)
+\frac{\omega(x)}{k-1} +|\omega(x)|^{k-1}
\omega(x) =0, \quad x\in {\Bbb R}^n
$$
By assuming $\omega$ is radial, i.e., $\omega(x)=v(|x|)$ with $v: [0,
\infty)\longmapsto {\Bbb R}$, the equation is further reduced to
\begin{equation}\label{eq:3.4}
v''(x) +\left(\frac{n-1}{x} +\frac{x}{2}\right)v'(x)
+\frac{v(x)}{k-1} +|v(x)|^{k-1}v(x)=0,
\quad x>0
\end{equation}
\vspace{.14in}
Haraux and Weissler \cite{hw} consider the solutions of
Equation (\ref{eq:3.4}) and we need to
use the following result of theirs.
\begin{prop}\label{prop:3.1}
Let $k>1$ and $n\ge 1$. If
$$
1 <\frac{n(k-1)}{2}< k+1
$$
then for some $v_0$,
there is a unique solution $v\in C^2([0,
\infty))$ to Equation (\ref{eq:3.4}) with $v(0)=v_0$ and $v'(0)=0$
such that
$$
\lim_{x\to \infty}x^m v(x) =0,\quad \mbox{for all $m>0$}
$$
\end{prop}
\vspace{.2in}
To prove the theorem, we only need to prove the following assertions
about the solution $\Phi=t^{-\frac{1}{k-1}}v(\frac{|x|}{\sqrt{t}})$
constructed above.
\begin{prop}
Assume that the indices $k,n,p$ and $r$ satisfy (\ref{eq:3.3}). Then
\begin{description}
\item[(1)]
$$
\Phi(t)\to 0,\qquad \mbox{in}\quad S'({\Bbb R}^n)\quad\mbox{ as}\quad
t\to 0
$$
where $S'({\Bbb R}^n)$ is the space of tempered distributions.
\item[(2)]
$$
\Phi(t)\to 0, \qquad\mbox{in}\quad \dot{L}_{r,p}\quad\mbox{as}\quad t\to 0
$$
\end{description}
\end{prop}
{\bf Proof.}\quad
To prove assertion (1), we calculate
for any $\phi\in S$,
$$
\lim_{t\to 0}\int \Phi(x,t)\phi(x)dx =\lim_{t\to 0}\left( t^{-\frac{1}{k-1}}
\int v(\frac{|x|}{\sqrt{t}})\phi(x)dx\right)
$$
$$
\le \lim_{t\to 0}\left( t^{-\frac{1}{k-1}+\frac{n}{2p}}\|v\|_{L^p}
\|\phi\|_{L^q}\right),\quad \mbox{with} \quad \frac{1}{p}+\frac{1}{q}=1
$$
Since $\|v\|_{L^p}$ is finite as implied by Proposition \ref{prop:3.1},
we conclude
that the above limit is zero.
\vspace{.14in}
To prove (2), we need the following lemma.
\begin{lemma}\label{lem:3}
Let $q\in (1,\infty)$, $00$, there is a unique solution $u(t)$
of the IVP (\ref{eqk:1.1}),(\ref{eqk:1.2}) on the time interval $[0,T]$
satisfying
$$
u\in BC_s((0,T],H^r),\quad \mbox{for any $r\ge 0$}
$$
Furthermore, for any $T'\in (0,T)$, there exists a neighborhood $V$ of
$u_0$ in $H^{s}$ such that the mapping
$$
\Phi: V\longmapsto BC_s((0,T'], H^{r}),
$$
is Lipschitz.
\end{thm}
\begin{rem}
The theorem remains unchanged if the nonlinear term $u^2$
in Equation (\ref{eqk:1.1}) is replaced by
$-u^2$. At this point the nonlinear heat equation differs from the
nonlinear Schr\"{o}dinger equation for which the focusing and defocusing
cases are quite different (see e.g. Bourgain \cite{b}).
\end{rem}
The proof of this theorem is again based on the contraction mapping
principle. We write Equation (\ref{eqk:1.1}) in the integral form
$$
u=U (u_0)(t)+G(u^2)(t)\equiv e^{-\Delta t}u_0 +\int_{0}^{t}
e^{-\Delta(t-\tau)}(u^2)(\tau) d\tau
$$
Then we estimate the operators $U$ and $G$ on $BC_s((0,T],H^r)$. The main
estimates are established in the propositions that follow.
\vspace{.16in}
\begin{prop}
\label{prop21}
Let $00$, $\Phi$ maps $X_{T,R}$ into $X_{T,R}$ and is a
contraction. Thus there is a unique fixed point $u=\Phi(u)$ in $X_{T,R}$.
It is clear that by reducing the time interval $(0,T)$, we can
extend the uniqueness to $X_{T,R'}$ for any $R'$ and thus to the whole
class in $X_T$.
\vspace{.1in}
For $T'\in (0,T)$, we can see from (\ref{tk})
that $K$ can be replaced by a larger
$K'$ such that (\ref{tk}) still holds for $K'$ and $T'$. That is, $\Phi$ is still
a contraction map for $v_0\in V$ where $V$ is some neighborhood of $u_0$.
The Lipschitz continuity of $\Phi$ is easily obtained by using tha fact
that $\Phi$ is contraction map on $V$. This finishes the proof of this
theorem.
\vspace{.22in}
Now we state a lemma which will be used
in the proof of Proposition \ref{prop22}.
In what follows we will
denote $(1+|\xi|^2)^{1/2}$ by $w(\xi)$ where $\xi \in {\Bbb R}^n$.
\begin{lemma}
\label{lem}
Let $r\ge 0,a$ be real numbers. If $g,h \in H^r$, then
$$
\|w(\xi a)^r\widehat{gh}(\xi)\|_{L^\infty}\le
\|w(\xi a)^r\hat{g}(\xi)\|_{L^2}\|w(\xi a)^r\hat{h}(\xi)\|_{L^2}
$$
\end{lemma}
The proof of this lemma is simple and can be found in \cite{dix1}.
\vspace{.22in}
\noindent{\bf Proof of Proposition \ref{prop22}}. \quad
First we estimate $\|u_G\|_{BC_s((0,T], H^q)}$.
We only need to prove for the case $r\le q\le r+2-\frac{n}{2}$ since
the norm is a nondecreasing function of $q$.
It is easy to check that
for $00, b>0$ the Beta function
$$
B(a,b)=\int_{0}^{1}(1-x)^{a-1} x^{b-1} dx
$$
is finite.
$II$ can be estimated in a quite similar way and the final result is the
same as that of $I$ apart from the constant $C$ may be different.
\vspace{.12in}
Now we show that $u_G: (0,T]\to H^q$ is continuous. Let $t_1,t_2\in (0,T]$
and we estimate the difference
$$
\|u_G(t_2)-u_G(t_1)\|_{H^q}
\le\left\|w(\xi)^q\int_{t_1}^{t_2}e^{-|\xi|^2(t_2-\tau)}
\widehat{u^2}(\tau)(\xi)
d\tau\right\|_{L^2}
$$ $$
+\left\|w(\xi)^q\int_{0}^{t_1}\left[e^{-|\xi|^2(t_2-\tau)}-e^{-|\xi|^2
(t_1-\tau)}\right]\widehat{u^2}(\tau)(\xi)d\tau \right\|_{L^2}
=III+IV
$$
In a similar manner $III$ and $IV$ can be estimated and consequently
we can show that for $s\le q\le r+2-\frac{n}{2}$:
$$
III,\quad IV \to 0,\quad \mbox{as} \quad t_2-t_1\quad \to 0
$$
\vspace{.12in}
Finally we show that
$$
u_G(t)\to 0,\quad \mbox{in $H^s$ as $t\to 0$}
$$
We estimate
$$
\|u_G\|_{H^s}=\left\|w(\xi)^s\int_{0}^{t}e^{-|\xi|^2(t-\tau)}\widehat{u^2}
(\tau)(\xi)d\tau\right\|_{L^2}
$$
and this can be done similarly as before. We omit details.
\newpage
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\end{document}