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论文类型：期刊论文

发表时间：2008-12-01

发表刊物：JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

收录刊物：SCIE、Scopus

卷号：348

期号：1

页面范围：234-243

ISSN号：0022-247X

关键字：parabolic equation of fourth order; existence of solutions; semi-discretization asymptotic behavior; large-time behavior; entropy method

摘要：This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: u(t) +del . (vertical bar del Delta u vertical bar(p-2)del Delta u) = f(u) in ohm R-N with boundary condition u = Delta u = 0 and initial data u(0). The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u(0) as p -> infinity. (C) 2008 Elsevier Inc. All rights reserved.