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Indexed by:期刊论文

Date of Publication:2017-02-01

Journal:ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES

Included Journals:SCIE、Scopus

Volume:33

Issue:1

Page Number:63-68

ISSN No.:0168-9673

Key Words:variable source; Fujita type condition; heat equation; global solutions; blow-up

Abstract:This paper studies heat equation with variable exponent ut triangle.u `vertical bar u(p(x)) + u(q) in R-N x ( 0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p- = inf p(x) = infp(x) <= p(X) <= sup p( x) = p+. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max {p+, q} <= 1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 < q = 1 with p+ > 1, or 1 < q < 1+ 2\N. In addition, if q > 1+ 2/N, then (i) all solutions blow up in finite time with 0 < p- = p+ = 1 + 2 N; (ii) there are both global and nonglobal solutions for p- > 1 + 2/N; and (iii) there are functions p(x) such that all solutions blow up in finite time, and also functions p(x) such that the problem possesses global solutions when p- < 1+ 2/N < p+.