Blow-up phenomenon to the semilinear heat equation for unbounded Laplacians on graphs

Abstract

Let \(G=(V,E)\) be an infinite graph. The purpose of this paper is to investigate the nonexistence of global solutions for the following semilinear heat equation

$$\begin{aligned} \left\{ \begin{array}{lc} \partial _t u=\Delta u + u^{1+\alpha }, &{}\, t>0,x\in V,\\ u(0,x)=u_0(x), &{}\, x \in V, \end{array} \right. \end{aligned}$$

where \(\Delta \) is an unbounded Laplacian on G, \(\alpha \) is a positive parameter and \(u_0\) is a nonnegative and nontrivial initial value. Using on-diagonal lower heat kernel bounds, we prove that the semilinear heat equation admits the blow-up solutions, which is viewed as a discrete analog of that of Fujita (J Fac Sci Univ Tokyo 13:109–124, 1966) and had been generalized to locally finite graphs with bounded Laplacians by Lin and Wu (Calc Var Partial Diff Equ 56(4):22, 2017). In this paper, new techniques have been developed to deal with unbounded graph Laplacians.