Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory
type: Item
instance of: scholarly article
title: Uniform decay of solutions for von Karman equations of dynamic viscoelasticity with memory
zbMATH Open document ID: 1197.35056
author: Jong Yeoul Park
author: Jum-Ran Kang
published in: Acta Applicandae Mathematicae
publication date: 2010-05-19T00:00:00Z
review text: Von Karman plates are thin structures with linear constitutive relations involving geometric nonlinearities arising from large rotations although strains are small. In this work, a linear viscoelastic constitutive relation is adopted with a given relaxation function. Hence equations describing the motion of the plate becomes nonlinear integro-differential equations because of the hereditary integral due to the viscoelastic term. In addition, nonlinear dissipative boundary conditions are prescribed on some part the sufficiently smooth boundary of the plate. The complementary part of the boundary is clamped. The authors first prove the existence of the solution for such a plate. To this end they employ the well known Galerkin method involving first a finite expansion. Imposing some restriction on the behaviour of the solution and making use of properties of Hilbert and some Sobolev spaces they manage to show uniqueness by a limiting process. Furthermore, they prove that an appropriately defined energy functional decays to zero with time if the relaxation function decreases exponentially.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 74K20
Mathematics Subject Classification ID: 74D05
Mathematics Subject Classification ID: 74H20
Mathematics Subject Classification ID: 74H40
Mathematics Subject Classification ID: 65M60
zbMATH DE Number: 5709156
zbMATH Keywords: von Karman system
zbMATH Keywords: boundary dissipation
zbMATH Keywords: exponential decay
zbMATH Keywords: Galerkin approximation
zbMATH Keywords: geometric nonlinearities
zbMATH Keywords: nonlinear integro-differential equations
zbMATH Keywords: nonlinear boundary conditions
reviewed by: Erdoğan S. Şuhubi
MaRDI profile type: Publication
full work available at URL: s10440-009-9520-7
OpenAlex ID: W2052913944
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DOI: 10.1007/S10440-009-9520-7
description: scientific article; zbMATH DE number 5709156