Semilinear equations

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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, 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

  1. 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 
  2. Cabre, X.; Sire, Yannick (2010), "Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates", Arxiv preprint arXiv:1012.0867 
  3. 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 
  4. 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 
  5. Felmer, P.; Quaas, A.; Tan, J., Positive Solutions Of Nonlinear Schrodinger Equation With The Fractional Laplacian. 
  6. 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 
  7. Palatucci, G.; Valdinoci, E.; Savin, O. (2011), "Local and global minimizers for a variational energy involving a fractional norm", Arxiv preprint arXiv:1104.1725 
  8. 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 
  9. 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 
  10. 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. 11.0 11.1 Kiselev, A. (to appear), "Nonlocal maximum principles for active scalars", Advances in Mathematics 
  12. 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. 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 
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