The existence and nonexistence of global solutions for a semilinear heat equation on graphs

Let G = (V, E) be a finite or locally finite connected weighted graph, Delta be the usual graph Laplacian. Using heat kernel estimates, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G {ut = Delta u + u(1+alpha) in (0,+infinity) x V, u(0, x) = a(x) in V. We conclude that, for a graph satisfying curvature dimension condition CDE'(n, 0) and V(x, r) similar or equal to r(m), if 0 < m alpha < 2, then the non-negative solution u is not global, and if m alpha > 2, then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results apply to the lattice Z(m).