Large time behavior of solutions to elastic wave with structural damping
type: Item
instance of: scholarly article
title: Large time behavior of solutions to elastic wave with structural damping
zbMATH Open document ID: 1492.35160
author: H. Takeda
published in: Journal of Differential Equations
publication date: 2022-05-04T00:00:00Z
review text: In the present paper the author studies the following linear Cauchy problem for elastic wave equations with different structurally damping terms: \begin{align*} & u_{tt}-\mu \Delta u-(\mu+\lambda) \nabla (\operatorname{div} u) + B u_t=0,\,\,\,(t,x)\in (0,\infty)\times \mathbb{R}^n,\\ & \quad u(0,x)=u_0(x), \,\,\,u_t(0,x)=u_1(x),\,\,\, x \in \mathbb{R}^n. \end{align*} Here, \(B u_t\) is either a structurally damped term \(\nu(-\Delta)^\sigma u_t\) or a structurally damped term of Kelvin-Voigt type \(\nu (-\mu \Delta -(\mu+\lambda) \nabla \operatorname{div})^\sigma u_t\) with \(\nu>0\) and \(\sigma \in (0,1]\). The data are supposed to belong to an energy space (even of higher order) with additional \(L^1\) regularity. The main goals of the author are to derive sharp decay estimates of the energies even of higher order and to show the optimality of these estimates by getting estimates from below, too. Moreover, the asymptotic profile of energy solutions even with suitable additional regularity is obtained by using Fourier multipliers connected with classical structurally damped wave equations. The main tools to prove all these results are a careful analysis in the Fourier space, some explicit representations of solutions to classical structurally damped waves, treatment of Fourier multipliers, oscillating integrals, respectively and the theory of special functions. It would be nice to have results (blow-up or global (in time) existence of small data Sobolev solutions) for related semilinear models with power nonlinearities.
Mathematics Subject Classification ID: 35L52
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 7517624
zbMATH Keywords: elastic waves
zbMATH Keywords: structural damping
zbMATH Keywords: Kelvin-Voigt damping
zbMATH Keywords: energy estimates
zbMATH Keywords: asymptotic profile
zbMATH Keywords: lower bounds
zbMATH Keywords: Fourier space analysis
reviewed by: Michael Reissig
MaRDI profile type: Publication
full work available at URL: j.jde.2022.04.017
OpenAlex ID: W4224254787
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DOI: 10.1016/J.JDE.2022.04.017
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description: scientific article