Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order
type: Item
instance of: scholarly article
title: Existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order
zbMATH Open document ID: 1176.35095
author: Bo Liang
author: Sining Zheng
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-10-06T00:00:00Z
review text: In this work the existence of weak solutions of the parabolic partial differential equation \(u_t+ \nabla\cdot (|\nabla \Delta u|^{p-2}\nabla\Delta u)= f(u)\) with boundary conditions \(u=\Delta u=0\) and initial condition \(u_0\) is analyzed. For the stationary solutions, a functional minimization and Leray-Schauder's fixed point theorem lead to the existence statement. For the developing case semi-discrete problems are solved and the solutions are used to define a second approximate solution. Both tend in the limit to the sought unique weak solution. This is shown by determination suitable bounds on the derivatives in time and space, and using a compactness argument. In the end of this work it is shown that in the \(p\to \infty\) limit the solutions converge to the initial data.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K35
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 5349083
zbMATH Keywords: entropy method
zbMATH Keywords: semi-discretization
zbMATH Keywords: stationary solutions
zbMATH Keywords: Leray-Schauder's fixed point theorem
zbMATH Keywords: compactness argument
reviewed by: M. D. Korzec
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.07.022
OpenAlex ID: W2033997732
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DOI: 10.1016/J.JMAA.2008.07.022
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description: scientific article; zbMATH DE number 5349083