Rational decay rates for a PDE heat--structure interaction:   Afrequency domain approach

George Avalos; Roberto Triggiani

doi:10.3934/eect.2013.2.233

Article Contents

2013, Volume 2, Issue 2: 233-253. Doi: 10.3934/eect.2013.2.233

This issue Previous Article Stability analysis of non-linear plates coupled with Darcy flows Next Article Regular solutions of wave equations with super-critical sources and exponential-to-logarithmic damping

Rational decay rates for a PDE heat--structure interaction: A frequency domain approach

George Avalos^1, and
Roberto Triggiani^2,

1.
Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588
2.
Department of Mathematics, University of Virginia, Charlottesville, VA 22903

Received: November 30, 2012

Revised: January 31, 2013

Published: May 2013

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Abstract

In this paper, we consider a simplified version of a fluid--structure PDE model ---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}}) $ (see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.

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Mathematics Subject Classification: Primary: 35M13, 93D20.

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