Anisotropic nonlocal diffusion equations with singular forcing

TL;DR

This paper establishes the mathematical foundations for anisotropic nonlocal heat equations with irregular forcing and singular spectral measures, focusing on existence, uniqueness, and regularity of solutions.

Contribution

It provides new existence, uniqueness, and regularity results for anisotropic nonlocal heat equations with minimal regularity assumptions and singular spectral measures.

Findings

Proved existence and uniqueness of solutions.

Established regularity properties of solutions.

Developed heat kernel estimates for anisotropic operators.

Abstract

We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very few regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.