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Extinction and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity

Yanchao Gao* and Wenjie Gao

Author Affiliations

School of Mathematics, Jilin University, Changchun, 130012, P.R. China

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Boundary Value Problems 2013, 2013:164  doi:10.1186/1687-2770-2013-164

Published: 10 July 2013

Abstract

The aim of this paper is to study the existence and extinction of weak solutions of the initial and boundary value problem for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M1">View MathML</a>. First, the authors apply the method of parabolic regularization and Galerkin’s method to prove the existence of solutions to the problem mentioned and then obtain the comparison principle by arguing by contradiction. Furthermore, the authors prove that the solution vanishes in finite time and approaches 0 in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M2">View MathML</a> norm as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M3">View MathML</a>.

Keywords:
nonlinear parabolic equation; nonstandard growth condition; extinction; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/164/mathml/M4">View MathML</a>-Laplace operator