Nonlinear nonlocal reaction-diffusion problem with local reaction

TL;DR

This paper investigates the long-term behavior of nonlinear nonlocal reaction-diffusion equations in metric measure spaces, establishing conditions for global existence, stability, and asymptotic bounds of solutions, especially for logistic nonlinearities.

Contribution

It introduces new analysis techniques for nonlocal diffusion problems with local reactions, including maximum principles, extremal equilibria, and stability criteria, highlighting differences from local diffusion models.

Findings

Solutions are globally defined for certain nonlinearities.

Maximum and comparison principles hold for solutions.

Existence of unique globally stable positive stationary solutions.

Abstract

In this paper we analyse the asymptotic behaviour of some nonlocal diffusion problems with local reaction term in general metric measure spaces. We find certain classes of nonlinear terms, including logistic type terms, for which solutions are globally defined with initial data in Lebesgue spaces. We prove solutions satisfy maximum and comparison principles and give sign conditions to ensure global asymptotic bounds for large times. We also prove that these problems possess extremal ordered equilibria and solutions, asymptotically, enter in between these equilibria. Finally we give conditions for a unique positive stationary solution that is globally asymptotically stable for nonnegative initial data. A detailed analysis is performed for logistic type nonlinearities. As the model we consider here lack of smoothing effect, important focus is payed along the whole paper on differences in the results with respect to problems with local diffusion, like the Laplacian operator.