Well-posedness of an evolution problem with nonlocal diffusion

TL;DR

This paper establishes the well-posedness of a general evolution problem involving nonlocal diffusion, under different assumptions on kernels and initial data, with applications in image processing and population dynamics.

Contribution

It introduces new conditions for well-posedness of nonlocal diffusion problems, relaxing regularity assumptions on kernels and initial data compared to previous work.

Findings

Proves well-posedness under Lipschitz and bounded variation assumptions.

Extends results to H"older continuous kernels with monotonicity.

Applicable to models in image processing and population dynamics.

Abstract

We prove the well-posedness of a general evolution reaction-nonlocal diffusion problem under two sets of assumptions. In the first set, the main hypothesis is the Lipschitz continuity of the range kernel and the bounded variation of the spatial kernel and the initial datum. In the second set of assumptions, we relax the Lipschitz continuity of the range kernel to H\"older continuity, and assume monotonic behavior. In this case, the spatial kernel and the initial data can be just integrable functions. The main applications of this model are related to the fields of Image Processing and Population Dynamics.