On nonexistence of global solutions for a semilinear heat equation on graphs

Let G=(V,E) be a simple, finite, connected, weighted graph satisfying curvature condition CDE′(n,0) and polynomial volume growth V(x,r)≤c0rm, Δη be the normalized Laplacian. In this paper we prove that the semilinear heat equation ut=Δηu+u1+α on G has no non-negative global solutions for any bounded, non-negative and non-trivial initial value in the case of mα=2. The obtained result provides a significant complement to the work that was done recently by Lin and Wu (2017) concerning the existence and nonexistence of global solutions for the semilinear heat equation on graphs.