BLOW-UP PROBLEMS FOR NONLINEAR PARABOLIC EQUATIONS ON LOCALLY FINITE GRAPHS

摘要

Let G =(V, E) be a locally finite connected weighted graph, and ? be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ?u + f(u) on G. The blow-up phenomenons for ut = ?u + f(u) 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 corresponding solutions will blow up in a finite time.