Nonlocal doubly nonlinear diffusion problems with nonlinear boundary conditions

TL;DR

This paper investigates the existence and uniqueness of solutions for nonlocal nonlinear diffusion problems with complex boundary conditions, encompassing various classical diffusion models within a unified nonlocal framework.

Contribution

It introduces a comprehensive analysis of nonlocal p-Laplacian type problems with nonlinear boundaries in metric random walk spaces, extending classical diffusion theories.

Findings

Established existence and uniqueness of solutions

Unified treatment of various classical diffusion problems

Extended analysis to nonlinear dynamical boundary conditions

Abstract

We study the existence and uniqueness of mild and strong solutions of nonlocal nonlinear diffusion problems of $p$-Laplacian type with nonlinear boundary conditions posed in metric random walk spaces. These spaces include, among others, weighted discrete graphs and $\mathbb{R}^N$ with a random walk induced by a nonsingular kernel. We also study the case of nonlinear dynamical boundary conditions. The generality of the nonlinearities considered allow us to cover the nonlocal counterparts of a large scope of local diffusion problems: Stefan problems, Hele-Shaw problems, diffusion in porous media problems, obstacle problems, and more. Nonlinear semigroup theory is the basis for this study.