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Indexed by:期刊论文

Date of Publication:2013-09-15

Journal:JOURNAL OF FUNCTIONAL ANALYSIS

Included Journals:SCIE、Scopus

Volume:265

Issue:6

Page Number:941-952

ISSN No.:0022-1236

Key Words:Parabolic system; Bounded domain; Critical Fujita exponent; Heat semigroup

Abstract:This paper considers the time-weighted parabolic system u(t) = Delta u + e(alpha t)upsilon(p), upsilon(t) = Delta upsilon + e(beta t)u(q) in bounded domain with alpha, beta is an element of R and p,q > 0, subject to null Dirichlet boundary condition. The critical Fujita curve is determined as (pq)(c) = 1+max{alpha+beta p,beta+alpha q,0}/lambda(1), where lambda(1) is the first eigenvalue of the Laplacian. As an extension, it is observed for another coupled system U-t = Delta U + mU + V-p, V-t = Delta V + nV + U-q with pq > 1 that there is the Fujita critical coefficient max{m, n} = lambda(1), namely, any nontrivial solution blows up in finite time if and only if max{m, n} >= lambda(1). The studies of critical curves for coupled systems in the current literature are all heavenly relying upon Jensen's inequality and the Kaplan method; for which one has to deal with complicated discussions on the exponents p, q being greater or less than 1. Differently, in the present framework, the heat semigroup is introduced to study critical curves for coupled systems, where various superlinear and sublinear cases can be treated uniformly by estimates involved. This greatly simplifies the arguments for establishing Fujita type theorems. (C) 2013 Elsevier Inc. All rights reserved.