%0 Journal Article
%T Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations
%A Yusuke Yamauchi
%J International Journal of Differential Equations
%D 2012
%I Hindawi Publishing Corporation
%R 10.1155/2012/417261
%X Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author. 1. Introduction 1.1. Fujita Type Results We first recall the result on the Cauchy problem for a semilinear heat equation: where , is the -dimensional Laplacian, and . Let be a bounded continuous function on . In pioneer work [1], Fujita showed that the exponent plays the crucial role for the existence and nonexistence of the solutions of (1.1). Let denote the Gaussian heat kernel: . Theorem 1.1 (see [1]). Suppose that and that its all derivatives are bounded.(i)Let . Then there is no global solution of (1.1) satisfying that£¿£¿ for£¿£¿ and .(ii)Let . Then for any there exists with the following property: if then there exists a global solution of (1.1) satisfying for and . In [2], Hayakawa showed first that there is no global solution of (1.1) in the critical case when or 2. Theorem 1.2 (see [2]). In case of , or , , (1.1) has no global solutions for any nontrivial initial data. In genaral space dimensions, Kobayashi et al. [3] consider the following problem: where and . Let be a bounded continuous function on . Theorem 1.3 (see [3]). Suppose that satisfies the following three conditions:(a) is a locally Lipschitz continuous and nondecreasing function in with£¿£¿ and for ,(b) for some ,(c)there exists a positive constant such that Then each positive solution of (1.5) blows up in finite time. Remark 1.4. (i) We remark that the proofs of the theorems in [2, 3] are mainly based on the iterated estimate from below obtained by the following integral equation: (ii) The critical nonlinearity of power type satisfies the assumptions (a), (b), and (c) in [3]. Weissler proved the nonexistence of global solution in -framework in [4]. The proof is quite short and elegant. Theorem 1.5 (see [4]). Suppose and that in £¿ is not identically zero. Then there is no nonnegative global solution to the integral (1.7) with initial value . The outline of the proof is as follows. First we assume that there is a global solution. From the fact that the solution for some , we can obtain that . This contradicts the boundedness of for large . Hence the solution is not global. Existence and nonexistence results for time-global solutions of (1.1) are summarized as follows.(i)Let . Then every nontrivial solution of (1.1) blows up in finite time.(ii)Let . Then (1.1) has a time-global
%U http://www.hindawi.com/journals/ijde/2012/417261/