Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let $G=(V,E)$ be a locally finite connected weighted graph, $\mathrm{\Delta }$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\mathrm{\Delta }u+f(u)$ on $G$. The blow-up phenomenons of the equation are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if $f$ satisfies appropriate conditions, then the solution of the equation blows up in a finite time.