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

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Let G=(V,E) be a finite or locally finite connected weighted graph, Δ be the usual graph Laplacian. Using heat kernel estimate, we prove the existence and nonexistence of global solutions for the following semilinear heat equation on G equation* { arraylc uₜ=Δu + u¹⁺α & in (0,+∞)× V, u(0,x)=a(x) & in V. array. equation* We conclude that, for a graph satisfying curvature dimension condition CDE'(n,0) and V(x,r)≃ rᵐ, if 0<mα<2, then the non-negative solution u is not global, and if mα>2, then there is a non-negative global solution u provided that the initial value is small enough. In particular, these results are true on lattice Zᵐ.