Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains

TL;DR

This paper studies the long-term dynamics of reaction-diffusion equations with nonlocal boundary conditions on irregular, fractal-like domains, revealing new insights into attractor behavior in non-smooth geometries.

Contribution

It introduces a general framework for analyzing reaction-diffusion equations with fractional boundary diffusion on rough domains, extending existing results to irregular geometries.

Findings

Existence of finite-dimensional global and exponential attractors.

Applicability to domains with Holder continuous and fractal boundaries.

Extension of classical reaction-diffusion results to non-smooth settings.

Abstract

We investigate the long term behavior in terms of finite dimensional global and exponential attractors, as time goes to infinity, of solutions to a semilinear reaction-diffusion equation on non-smooth domains subject to nonlocal Robin boundary conditions, characterized by the presence of fractional diffusion on the boundary. Our results are of general character and apply to a large class of irregular domains, including domains whose boundary is Holder continuous and domains which have fractal-like geometry. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, for the well-known reaction-diffusion equation in smooth domains, the framework we develop also makes possible a number of new results for all diffusion models in other non-smooth settings.