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

Let \(G=(V,E)\) $G=(V,E)$ be an infinite graph. The purpose of this paper is to investigate the nonexistence of global solutions for the following semilinear heat equation \(\begin{aligned} \left\{ \begin{array}{lc} \partial _t u=\Delta u + u^{1+\alpha }, &{}\, t>0,x\in V,\\ u(0,x)=u_0(x), &{}\, x \in V, \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\partial }_{t}u=\mathrm{\Delta }u+u^{1+\alpha },\hfill & t>0,x\in V,\\ u(0,x)=u_0(x),\hfill & x\in V,\end{array}\right)\end{array}$$ where \(\Delta \) $\mathrm{\Delta }$ is an unbounded Laplacian on G, \(\alpha \) $\alpha$ is a positive parameter and \(u_0\) $u_0$ 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.