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

Abstract

Let $G=(V,E)$ be a finite or locally finite connected weighted graph, $\Delta$ 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 $(0,+\infty)\times V$,}\\ u(0,x)=a(x) &\, \text{in $V$.} \end{array} \right. \end{equation*} We conclude that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ 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 $\mathbb{Z}^m$.