Semilinear equations

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 $s<1/2$.

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 .

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 , 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 $s<1/2$ . Still, if $s \in (0,1/2)$, the solution regularizes for large enough times.

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 $s > 1/2$ . 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) or the modulus of continuity approach .

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. 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.