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

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