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

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 $hv+(-\Delta {)}^{\sigma /2}v=f$, $h>0$, 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 $\sigma =2$.

42 pages, 1 figure

References in corpus (1)

• From the long jump random walk to the fractional Laplacian

Cited by in corpus (1)

• The Fisher-KPP equation with nonlinear fractional diffusion