Convergence of solutions of a rescaled evolution nonlocal cross-diffusion problem to its local diffusion counterpart

TL;DR

This paper demonstrates that solutions of a rescaled nonlocal cross-diffusion problem converge to those of the local diffusion problem, providing a new proof of existence for the local case.

Contribution

It establishes the convergence of nonlocal solutions to local solutions under kernel rescaling, offering a novel proof of existence for the local diffusion problem.

Findings

Solutions of nonlocal cross-diffusion converge to local diffusion solutions

Rescaling the kernel leads to the local diffusion limit

Provides a new proof of existence for local diffusion solutions

Abstract

We prove that, under a suitable rescaling of the integrable kernel defining the nonlocal diffusion terms, the corresponding sequence of solutions of the Shigesada-Kawasaki-Teramoto nonlocal cross-diffusion problem converges to a solution of the usual problem with local diffusion. In particular, the result may be regarded as a new proof of existence of solutions for the local diffusion problem.