On a semilinear parabolic equation with time-dependent source term on infinite graphs

TL;DR

This work analyzes the Cauchy problem $u_t=\Delta u+h(t)u^q$ for $q>1$ on an infinite weighted graph with $\lambda_1(G)>0$, establishing a sharp dichotomy between global existence and finite-time blow-up as a function of the growth rate of $h(t)$. The authors combine heat-kernel techniques on graphs with fixed-point methods to obtain local and global mild solutions, and they derive a Jensen-type differential inequality to prove blow-up in the supercritical regime. A key contribution is the identification of a threshold $\alpha=(q-1)\lambda_1(G)$ for the exponential source $h(t)=e^{\alpha t}$: solutions blow up for $\alpha>(q-1)\lambda_1(G)$ and exist globally for $\alpha<(q-1)\lambda_1(G)$ with small initial data; the critical case remains open. The results extend blow-up/global-existence theories from hyperbolic spaces and Euclidean settings to infinite graphs, providing a robust framework for nonlinear diffusion with time-dependent sources on networks.

Abstract

We are concerned with semilinear parabolic equations, with a time-dependent source term of the form $h(t)u^q$ with $q>1$, posed on an infinite graph. We assume that the bottom of the $L^2$-spectrum of the Laplacian on the graph, denoted by $λ_1(G)$, is positive. In dependence of $q, h(t)$ and $λ_1(G)$, we show global in time existence or finite time blow-up of solutions.

Theorems & Definitions (27)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Definition 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 3.5