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, $\mathrm{\Delta }$ 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 $G$ \begin{equation*} \left{ \begin{array}{lc} u_t=\Delta u + u^{1+\alpha} &, $\text{in }(0,+\mathrm{\infty })\times V\text{,}$\ u(0,x)=a(x) &, \text{in $V$.} \end{array} \right. \end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CD{E}^{\mathrm{\prime }}(n,0)$ and $V(x,r)\simeq r^m$, 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 are true on lattice ${\mathbb{Z}}^m$.