# Semilinear equations

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== Some common semilinear equations == | == Some common semilinear equations == | ||

- | === | + | === Stationary equations with zeroth order nonlinearity === |

- | Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers | + | Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs. |

\[ (-\Delta)^s u = f(u). \] | \[ (-\Delta)^s u = f(u). \] | ||

- | If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^ | + | If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard [[bootstrapping]]. |

- | Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, | + | Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, boundedness, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained <ref name="OLC"/> <ref name="CS"/> <ref name="CC"/> <ref name="LF"/> <ref name="FQT"/> <ref name="SV"/> <ref name="PSV"/>. |

=== Reaction diffusion equations === | === Reaction diffusion equations === | ||

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\[ u_t + (-\Delta)^s u = f(u). \] | \[ u_t + (-\Delta)^s u = f(u). \] | ||

- | The case $f(u) = u(1-u)$ corresponds to the Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity. | + | The case $f(u) = u(1-u)$ corresponds to the KPP/Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior <ref name="CR"/>, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity. |

=== Burgers equation with fractional diffusion === | === Burgers equation with fractional diffusion === | ||

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where $u = R^\perp \theta$ (and $R$ is the Riesz transform). | where $u = R^\perp \theta$ (and $R$ is the Riesz transform). | ||

- | The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. | + | The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation. |

=== Conservation laws with fractional diffusion === | === Conservation laws with fractional diffusion === | ||

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\[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] | \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\] | ||

- | + | For Lipschitz initial data, the Cauchy problem always has a viscosity solution which is Lipschitz in space.<ref name="DI"/> The problem is well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$. | |

- | The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]]. The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]]. | + | The subcritical case $s>1/2$ can be solved with classical [[bootstrapping]].<ref name="DI"/> The critical case $s=1/2$ was solved using the regularity results for [[drift-diffusion equations]].<ref name="S"/> |

== References == | == References == | ||

{{reflist|refs= | {{reflist|refs= | ||

+ | <ref name="OLC">{{Citation | last1=Ou | first1=Biao | last2=Li | first2=Congming | last3=Chen | first3=Wenxiong | title=Classification of solutions for an integral equation | url=http://dx.doi.org/10.1002/cpa.20116 | doi=10.1002/cpa.20116 | year=2006 | journal=[[Communications on Pure and Applied Mathematics]] | issn=0010-3640 | volume=59 | issue=3 | pages=330–343}}</ref> | ||

<ref name="KNS">{{Citation | last1=Kiselev | first1=Alexander | last2=Nazarov | first2=Fedor | last3=Shterenberg | first3=Roman | title=Blow up and regularity for fractal Burgers equation | year=2008 | journal=Dynamics of Partial Differential Equations | issn=1548-159X | volume=5 | issue=3 | pages=211–240}}</ref> | <ref name="KNS">{{Citation | last1=Kiselev | first1=Alexander | last2=Nazarov | first2=Fedor | last3=Shterenberg | first3=Roman | title=Blow up and regularity for fractal Burgers equation | year=2008 | journal=Dynamics of Partial Differential Equations | issn=1548-159X | volume=5 | issue=3 | pages=211–240}}</ref> | ||

<ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | doi=10.1016/j.aim.2010.09.007 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=2020–2039}}</ref> | <ref name="S">{{Citation | last1=Silvestre | first1=Luis | title=On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion | url=http://dx.doi.org/10.1016/j.aim.2010.09.007 | doi=10.1016/j.aim.2010.09.007 | year=2011 | journal=Advances in Mathematics | issn=0001-8708 | volume=226 | issue=2 | pages=2020–2039}}</ref> | ||

<ref name="CCS">{{Citation | last1=Chan | first1=Chi Hin | last2=Czubak | first2=Magdalena | last3=Silvestre | first3=Luis | title=Eventual regularization of the slightly supercritical fractional Burgers equation | url=http://dx.doi.org/10.3934/dcds.2010.27.847 | doi=10.3934/dcds.2010.27.847 | year=2010 | journal=Discrete and Continuous Dynamical Systems. Series A | issn=1078-0947 | volume=27 | issue=2 | pages=847–861}}</ref> | <ref name="CCS">{{Citation | last1=Chan | first1=Chi Hin | last2=Czubak | first2=Magdalena | last3=Silvestre | first3=Luis | title=Eventual regularization of the slightly supercritical fractional Burgers equation | url=http://dx.doi.org/10.3934/dcds.2010.27.847 | doi=10.3934/dcds.2010.27.847 | year=2010 | journal=Discrete and Continuous Dynamical Systems. Series A | issn=1078-0947 | volume=27 | issue=2 | pages=847–861}}</ref> | ||

- | <ref name="K">{{Citation | last1=Kiselev | first1=A. | title= | + | <ref name="K">{{Citation | last1=Kiselev | first1=A. | title=Nonlocal maximum principles for active scalars | year=to appear | journal=Advances in Mathematics}}</ref> |

<ref name="DI">{{Citation | last1=Droniou | first1=Jérôme | last2=Imbert | first2=Cyril | title=Fractal first-order partial differential equations | url=http://dx.doi.org/10.1007/s00205-006-0429-2 | doi=10.1007/s00205-006-0429-2 | year=2006 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | volume=182 | issue=2 | pages=299–331}}</ref> | <ref name="DI">{{Citation | last1=Droniou | first1=Jérôme | last2=Imbert | first2=Cyril | title=Fractal first-order partial differential equations | url=http://dx.doi.org/10.1007/s00205-006-0429-2 | doi=10.1007/s00205-006-0429-2 | year=2006 | journal=Archive for Rational Mechanics and Analysis | issn=0003-9527 | volume=182 | issue=2 | pages=299–331}}</ref> | ||

+ | <ref name="CR">{{Citation | last1=Cabré | first1=Xavier | last2=Roquejoffre | first2=Jean-Michel | title=Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire | url=http://dx.doi.org/10.1016/j.crma.2009.10.012 | doi=10.1016/j.crma.2009.10.012 | year=2009 | journal=Comptes Rendus Mathématique. Académie des Sciences. Paris | issn=1631-073X | volume=347 | issue=23 | pages=1361–1366}}</ref> | ||

+ | <ref name="CS">{{Citation | last1=Cabre | first1=X. | last2=Sire | first2=Yannick | title=Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates | year=2010 | journal=Arxiv preprint arXiv:1012.0867}}</ref> | ||

+ | <ref name="CC"> {{Citation | last1=Cabré | first1=Xavier | last2=Cinti | first2=E. | title=Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian | year=2010 | journal=Discrete and Continuous Dynamical Systems (DCDS-A) | volume=28 | issue=3 | pages=1179–1206}} </ref> | ||

+ | <ref name="LF">{{Citation | last1=Frank | first1=R.L. | last2=Lenzmann | first2=E. | title=Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$ | year=2010 | journal=Arxiv preprint arXiv:1009.4042}}</ref> | ||

+ | <ref name="FQT">{{Citation | last1=Felmer | first1=P. | last2=Quaas | first2=A. | last3=Tan | first3=J. | title=Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian.}}</ref> | ||

+ | <ref name="SV"> {{Citation | last1=Sire | first1=Yannick | last2=Valdinoci | first2=E. | title=Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result | publisher=[[Elsevier]] | year=2009 | journal=Journal of Functional Analysis | issn=0022-1236 | volume=256 | issue=6 | pages=1842–1864}} </ref> | ||

+ | <ref name="PSV">{{Citation | last1=Palatucci | first1=G. | last2=Valdinoci | first2=E. | last3=Savin | first3=O. | title=Local and global minimizers for a variational energy involving a fractional norm | year=2011 | journal=Arxiv preprint arXiv:1104.1725}}</ref> | ||

}} | }} |

## Latest revision as of 19:27, 15 May 2015

An equation is called semilinear if it consists of the sum of a well understood linear term plus a lower order nonlinear term. For elliptic and parabolic equations, the two effective possibilities for the linear term is to be either the fractional Laplacian or the fractional heat equation.

Some equations which technically do not satisfy the definition above are still considered semilinear. For example evolution equations of the form \[ u_t + (-\Delta)^s u + H(x,u,Du) = 0 \] can be thought of as semilinear equations even if $s<1/2$.

## Contents |

## Some common semilinear equations

### Stationary equations with zeroth order nonlinearity

Adding a zeroth order term to the right hand side to either the Laplace equation or the fractional Laplace equation is probably the theme for which the largest number of papers have been written on PDEs. \[ (-\Delta)^s u = f(u). \] If $f$ is $C^\infty$ and some initial regularity can be shown to the solution $u$ (like $L^\infty$), then the solution $u$ will also be $C^\infty$, which can be shown by a standard bootstrapping.

Natural question to ask about this type of equations are about the existence of nontrivial global solutions that vanish at infinity, positivity of solutions, boundedness, symmetries, etc... Depending on the structure of the nonlinearity $f(u)$, different results are obtained ^{[1]} ^{[2]} ^{[3]} ^{[4]} ^{[5]} ^{[6]} ^{[7]}.

### Reaction diffusion equations

This general class refers to the equations we get by adding a zeroth order term to the right hand side of a heat equation. For the fractional case, it would look like \[ u_t + (-\Delta)^s u = f(u). \]

The case $f(u) = u(1-u)$ corresponds to the KPP/Fisher equation. For this and other related models, it makes sense to study solutions restricted to $0 \leq u \leq 1$. The research centers around traveling waves, their stability, limits, asymptotic behavior ^{[8]}, etc... Solutions are trivially $C^\infty$ so there is no issue about regularity.

### Burgers equation with fractional diffusion

It refers to the parabolic equation for a function on the real line $u:[0,+\infty) \times \R \to \R$,
\[ u_t + u \ u_x + (-\Delta)^s u = 0 \]
The equation is known to be well posed if $s \geq 1/2$ and to develop shocks if $s<1/2$ ^{[9]}. Still, if $s \in (0,1/2)$, the solution regularizes for large enough times^{[10]}^{[11]}.

### Surface quasi-geostrophic equation

It refers to the parabolic equation for a scalar function on the plane $\theta:[0,+\infty) \times \R^2 \to \R$, \[ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0 \] where $u = R^\perp \theta$ (and $R$ is the Riesz transform).

The equation is well posed if $s \geq 1/2$. The well posedness in the case $s < 1/2$ is a major open problem. It is believed that solving the supercritical SQG equation could possibly help understand 3D Navier-Stokes equation.

### Conservation laws with fractional diffusion

(aka "fractal conservation laws")
It refers to parabolic equations of the form
\[ u_t + \mathrm{div } F(u) + (-\Delta)^s u = 0.\]
The Cauchy problem is known to be well posed classically if $s > 1/2$ ^{[12]}. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The critical case $s=1/2$ appears not to be written anywhere. However, it can be solved following the same method as for the Hamilton-Jacobi equations with fractional diffusion (below) ^{[13]} or the modulus of continuity approach ^{[11]}.

### Hamilton-Jacobi equation with fractional diffusion

It refers to the parabolic equation \[ u_t + H(\nabla u) + (-\Delta)^s u = 0.\]

For Lipschitz initial data, the Cauchy problem always has a viscosity solution which is Lipschitz in space.^{[12]} The problem is well posed classically if $s \geq 1/2$. For $s<1/2$ there are viscosity solutions that are not $C^1$.

The subcritical case $s>1/2$ can be solved with classical bootstrapping.^{[12]} The critical case $s=1/2$ was solved using the regularity results for drift-diffusion equations.^{[13]}

## References

- ↑ Ou, Biao; Li, Congming; Chen, Wenxiong (2006), "Classification of solutions for an integral equation",
*Communications on Pure and Applied Mathematics***59**(3): 330–343, doi:10.1002/cpa.20116, ISSN 0010-3640, http://dx.doi.org/10.1002/cpa.20116 - ↑ Cabre, X.; Sire, Yannick (2010), "Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates",
*Arxiv preprint arXiv:1012.0867* - ↑ Cabré, Xavier; Cinti, E. (2010), "Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian",
*Discrete and Continuous Dynamical Systems (DCDS-A)***28**(3): 1179–1206 - ↑ Frank, R.L.; Lenzmann, E. (2010), "Uniqueness and Nondegeneracy of Ground States for $(-\Delta)^s Q+ Q-Q^{\alpha+1}= 0$ in $\R$",
*Arxiv preprint arXiv:1009.4042* - ↑ Felmer, P.; Quaas, A.; Tan, J.,
*Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian.* - ↑ Sire, Yannick; Valdinoci, E. (2009), "Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result",
*Journal of Functional Analysis*(Elsevier)**256**(6): 1842–1864, ISSN 0022-1236 - ↑ Palatucci, G.; Valdinoci, E.; Savin, O. (2011), "Local and global minimizers for a variational energy involving a fractional norm",
*Arxiv preprint arXiv:1104.1725* - ↑ Cabré, Xavier; Roquejoffre, Jean-Michel (2009), "Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire",
*Comptes Rendus Mathématique. Académie des Sciences. Paris***347**(23): 1361–1366, doi:10.1016/j.crma.2009.10.012, ISSN 1631-073X, http://dx.doi.org/10.1016/j.crma.2009.10.012 - ↑ Kiselev, Alexander; Nazarov, Fedor; Shterenberg, Roman (2008), "Blow up and regularity for fractal Burgers equation",
*Dynamics of Partial Differential Equations***5**(3): 211–240, ISSN 1548-159X - ↑ Chan, Chi Hin; Czubak, Magdalena; Silvestre, Luis (2010), "Eventual regularization of the slightly supercritical fractional Burgers equation",
*Discrete and Continuous Dynamical Systems. Series A***27**(2): 847–861, doi:10.3934/dcds.2010.27.847, ISSN 1078-0947, http://dx.doi.org/10.3934/dcds.2010.27.847 - ↑
^{11.0}^{11.1}Kiselev, A. (to appear), "Nonlocal maximum principles for active scalars",*Advances in Mathematics* - ↑
^{12.0}^{12.1}^{12.2}Droniou, Jérôme; Imbert, Cyril (2006), "Fractal first-order partial differential equations",*Archive for Rational Mechanics and Analysis***182**(2): 299–331, doi:10.1007/s00205-006-0429-2, ISSN 0003-9527, http://dx.doi.org/10.1007/s00205-006-0429-2 - ↑
^{13.0}^{13.1}Silvestre, Luis (2011), "On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion",*Advances in Mathematics***226**(2): 2020–2039, doi:10.1016/j.aim.2010.09.007, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2010.09.007