Coupled local/nonlocal models in thin domains

TL;DR

This paper studies coupled local and nonlocal diffusion models in thin domains, analyzing their behavior, existence, uniqueness, and qualitative properties, including the limit case where one domain becomes lower-dimensional.

Contribution

It introduces a coupled local/nonlocal diffusion model in thin domains and analyzes the limit where one domain reduces in dimension, establishing existence, uniqueness, and qualitative properties.

Findings

Proved existence and uniqueness of solutions.

Established conservation of mass.

Demonstrated convergence to initial mean value over time.

Abstract

In this paper, we analyze a model composed by coupled local and nonlocal diffusion equations acting in different subdomains. We consider the limit case when one of the subdomains is thin in one direction (it is concentrated to a domain of smaller dimension) and as a limit problem we obtain coupling between local and nonlocal equations acting in domains of different dimension. We find existence and uniqueness of solutions and we prove several qualitative properties (like conservation of mass and convergence to the mean value of the initial condition as time goes to infinity).