On existence and nonexistence global solutions of reaction-diffusion equations

We consider the initial value problem for the reaction-diffusion equation ut = ∆u + f(u). In this paper we show the existence and nonexistence of the global solutions in time. Especially, we extend the condition of the nonlinear terms to more general. We have the results of the existence and the nonexistence for the equation with the nonlinear term f satisfying lim infs→0 f(s)/sp > 0 and lim sups→0 f(s)/sq 0 and q > 0. Keyword and Phrases: reaction-diffusion, blow-up, global existence, nonlinear term. AMS subject classifications: 35K15, 35K57.