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Indexed by:期刊论文

Date of Publication:2009-10-01

Journal:JOURNAL OF DIFFERENTIAL EQUATIONS

Included Journals:SCIE、Scopus

Volume:247

Issue:7

Page Number:1980-1992

ISSN No.:0022-0396

Key Words:Critical exponent; Blow-up rate; Blow-up set; Spatial blow-up profile; Multi-nonlinear parabolic equation; Inner absorption; Gradient term; Nonlinear boundary flux

Abstract:This paper deals with parabolic equation u(t) = Delta u + vertical bar del u vertical bar(r) - ae(pu) subject to nonlinear boundary flux partial derivative u/partial derivative eta = e(qu), where r > 1, p, q, a > 0. There are two positive sources (the gradient reaction and the boundary flux) and a negative one (the absorption) in the model. It is well known that blow-up or not of solutions depends on which one dominating the model, the positive or negative sources, and furthermore on the absorption coefficient for the balance case of them. The aim of the paper is to study the influence of the reactive gradient term on the asymptotic behavior of solutions. We at first determine the critical blow-up exponent, and then obtain the blow-up rate, the blow-up set as well as the spatial blow-up profile for blow-up solutions in the one-dimensional case. It turns out that the gradient term makes a substantial contribution to the formation of blow-up if and only if r >= 2, where the critical r = 2 is such a balance situation of the two positive sources for which the effects of the gradient reaction and the boundary source are at the same level. In addition, it is observed that the gradient term with r > 2 significantly affects the blow-up rate also. In fact, the gained blow-up rates themselves contain the exponent r of the gradient term. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. As for the influence of gradient perturbations on spatial blow-up profiles, we only need some coefficients related to r for the profile estimates, while the exponent of the profile itself is r-independent. This seems natural for boundary blow-up solutions that the spatial profiles mainly rely on the exponent of the boundary singularity. (C) 2009 Elsevier Inc. All rights reserved.