期刊论文

Bai, Xueli

Zheng, SN (reprint author), Dalian Univ Technol, Sch Math Sci, Dalian 116024, Peoples R China.

Zheng, Sining,Wang, Wei

2013-09-15

JOURNAL OF FUNCTIONAL ANALYSIS

SCIE、Scopus

J

265

6

941-952

0022-1236

Parabolic system; Bounded domain; Critical Fujita exponent; Heat semigroup

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.