# Coupling local and nonlocal evolution equations

## TL;DR

This paper establishes the mathematical foundation for evolution equations combining local and nonlocal diffusion, proving existence, uniqueness, and qualitative properties, and exploring their long-term behavior and limiting cases.

## Contribution

It introduces a coupled local-nonlocal diffusion model, proving well-posedness and analyzing its properties, including mass preservation and asymptotic behavior.

## Key findings

- Proved existence and uniqueness of solutions.
- Demonstrated mass preservation for certain boundary conditions.
- Analyzed the large-time behavior and limiting cases to the classical heat equation.

## Abstract

We prove existence, uniqueness and several qualitative properties for evolution equations that combine local and nonlocal diffusion operators acting in different subdomains and coupled in such a way that the resulting evolution equation is the gradient flow of an energy functional. We deal with the Cauchy, Neumann and Dirichlet problems, in the last two cases with zero boundary data. For the first two problems we prove that the model preserves the total mass. We also study the behaviour of the solutions for large times. Finally, we show that we can recover the usual heat equation (local diffusion) in a limit procedure when we rescale the nonlocal kernel in a suitable way.