A note on the equations of a thermoelastic plate
type: Item
instance of: scholarly article
title: A note on the equations of a thermoelastic plate
zbMATH Open document ID: 0826.35048
DOI: 10.1016/0893-9659(95)00020-Q
author: Yanyan Li
published in: Applied Mathematics Letters
publication date: 1995-11-23T00:00:00Z
review text: We consider the equations \[ u_{tt} = - \Delta^2 u - \alpha \Delta \theta, \quad \theta_t = \beta \Delta \theta - \gamma \theta + \alpha \Delta u_t \] on a bounded domain \(\Omega \subset \mathbb{R}^2\) with a smooth boundary and subject to Dirichlet conditions \(u = {\partial u \over \partial n} = \theta = 0\), on \(\partial \Omega\). We prove that the equations are associated with an analytic semigroup. In particular, this implies exponential decay and further shows that the rate of decay is determined by the least stable eigenvalue. It also gives regularity properties of solutions which should be useful for studying nonlinear problems. We shall consider only the case of Dirichlet conditions, but the method should extend to other cases without major difficulty. One remarkable aspect of our result is that the analytic semigroup property is clearly introduced by the lower dimensional approximations which are involved in modeling plates and beams. The equations of thermoelasticity themselves do not generate an analytic semigroup.
Mathematics Subject Classification ID: 35K50
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 775961
zbMATH Keywords: thermoelastic plate
zbMATH Keywords: analytic semigroup
zbMATH Keywords: exponential decay
MaRDI profile type: Publication
cites work: Q3995611
cites work: On the Energy Decay of a Linear Thermoelastic Bar and Plate
cites work: On the exponential decay of the energy of a linear thermoelastic plate
full work available at URL: 0893-9659(95)00020-q
OpenAlex ID: W2075990920
Wikidata QID: Q127792683
description: scientific article; zbMATH DE number 775961