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

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 {partial derivative(t)u=Delta u+u(1+alpha),t>0, x is an element of V, u(0,x)=u(0) (x), x is an element of V, where Delta is an unbounded Laplacian on G, alpha is a positive parameter and u0 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.