Semilinear Diffusion Equations on Infinite Graphs: The Dissipative and Lipschitz Cases

TL;DR

This paper investigates semilinear diffusion equations on infinite graphs, establishing existence, uniqueness, and regularity of solutions for various nonlinearities using discretization and exhaustion techniques.

Contribution

It introduces a novel approach combining implicit Euler schemes and exhaustion methods to analyze solutions on infinite graphs under minimal assumptions.

Findings

Proved existence and uniqueness of solutions in ^p spaces.

Established finite-time extinction and positivity results.

Derived results for related time-independent equations.

Abstract

We study a class of semilinear diffusion equations on infinite, connected, weighted graphs, focusing on two types of nonlinearities: monotone decreasing and Lipschitz continuous. Under minimal structural assumptions on the graph, we establish existence, uniqueness, and regularity of mild solutions for initial data in $\ell^p$ spaces, with $1\leq p<\infty$. Our approach relies on time discretization via an implicit Euler scheme and an exhaustion technique using Dirichlet subgraphs. As a by-product, we obtain existence and uniqueness results for a related time-independent equation. Finite-time extinction and positivity for solutions under a specific forcing term are also proved.