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

TL;DR

This work develops a rigorous framework for semilinear diffusion equations on infinite graphs with two nonlinearities: monotone decreasing and uniformly Lipschitz $f$. It constructs unique mild solutions in $\ell^p(X,\mu)$ ($1\le p<\infty$) by time discretization with an implicit Euler scheme and an exhaustion by Dirichlet subgraphs, avoiding heat-kernel or spectral assumptions. A related time-independent equation is analyzed via accretivity on a dense set $\Omega_p$, yielding existence, uniqueness, and a priori estimates. In the dissipative setting, the authors prove a parabolic comparison principle, establish positivity results, and derive finite-time extinction and explicit barrier bounds for saturating nonlinearities $f(u)=-u|u|^{q-1}$, with sharp results in both linear and nonlinear regimes. The methods extend finite-graph results to infinite networks under minimal structural conditions, providing practical tools for nonlinear diffusion on graphs.

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.

Theorems & Definitions (48)

• Definition 2.1: Graph
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Lemma 2.6
• proof
• Proposition 2.7
• proof
• Definition 3.1: Classical Solution