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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$$begin{aligned} left{ begin{array}{lc} u_t=Delta u + u^{1+alpha } u0026{}, text {in }(0,+infty )times V, u(0,x)=a(x) u0026{}, text {in }V. end{array} right. end{aligned}$$We conclude that, for a graph satisfying curvature dimension condition (textit{CDE}u0027(n,0)) and (V(x,r)simeq r^m), if (0 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 ({mathbb {Z}}^m).

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