Nonlocal diffusion equations with dynamical boundary conditions

TL;DR

This paper investigates nonlocal diffusion equations with dynamical boundary conditions, establishing existence, uniqueness, and long-term behavior of solutions for both smooth and singular kernels.

Contribution

It introduces a framework for analyzing nonlocal problems with dynamical boundaries, extending classical results to nonlocal operators with various kernels.

Findings

Existence and uniqueness of solutions are proven.

Asymptotic behavior of solutions as time approaches infinity is characterized.

Results apply to both smooth and singular kernels.

Abstract

In this paper we study nonlocal problems that are analogous to the local ones given by the Laplacian or the p-Laplacian with dynamical boundary conditions. We deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity.