Symmetrization for Linear and Nonlinear Fractional Parabolic Equations of Porous Medium Type

TL;DR

This paper proves symmetrization results for linear and nonlinear fractional parabolic equations of porous medium type, extending classical results and providing counterexamples in certain nonlinear cases.

Contribution

It extends symmetrization techniques to fractional diffusion equations, including nonlinear cases with concave or convex nonlinearities, and constructs counterexamples for specific nonlinearities.

Findings

Symmetrization results hold for linear fractional diffusion equations.

Complete symmetrization is achieved for certain convex nonlinearities.

Counterexamples show limitations for convex nonlinearities in parabolic and elliptic cases.

Abstract

We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and itselliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of concentrations. The results extend to the nonlinear version, $\partial_t u+(-\Delta)^{\sigma/2}A(u)=f$, but only when $A:\re_+\to\re_+$ is a concave function. In the elliptic case, complete symmetrization results are proved for $\,B(v)+(-\Delta)^{\sigma/2}v=f$ \ when $B(v)$ is a convex nonnegative function for $v>0$ with $B(0)=0$, and partial results when $B$ is concave. Remarkable counterexamples are constructed for the parabolic equation when $A$ is convex, resp. for the elliptic equation when $B$ is concave. Such counterexamples do not exist in the standard diffusion case $\sigma=2$.