Nonlocal filtration equations with rough kernels

TL;DR

This paper investigates nonlinear nonlocal equations with rough kernels, establishing existence, uniqueness, regularity, and long-term behavior of solutions, especially for porous media type nonlinearities and kernels similar to fractional Laplacians.

Contribution

It introduces new existence and regularity results for solutions of nonlocal equations with singular kernels, extending the theory to rough kernels and nonlinearities like porous media.

Findings

Solutions are bounded and Hölder continuous for positive times.

Existence and uniqueness of weak solutions are proven.

Long-term behavior is characterized for kernels close to fractional Laplacians.

Abstract

We study the nonlinear and nonlocal Cauchy problem \[ \partial_{t}u+\mathcal{L}\varphi(u)=0 \quad\text{in }\mathbb{R}^{N}\times\mathbb{R}_+,\qquad u(\cdot,0)=u_0, \] where $\mathcal{L}$ is a L\'evy-type nonlocal operator with a kernel having a singularity at the origin as that of the fractional Laplacian. The nonlinearity $\varphi$ is nondecreasing and continuous, and the initial datum $u_0$ is assumed to be in $L^1(\mathbb{R}^N)$. We prove existence and uniqueness of weak solutions. For a wide class of nonlinearities, including the porous media case, $\varphi(u)=|u|^{m-1}u$, $m>1$, these solutions turn out to be bounded and H\"older continuous for $t>0$. We also describe the large time behaviour when the nonlinearity resembles a power for $u\approx 0$ and the kernel associated to $\mathcal{L}$ is close at infinity to that of the fractional Laplacian.