On the decay rate of local energy for the elastic wave equation
type: Item
instance of: scholarly article
title: On the decay rate of local energy for the elastic wave equation
zbMATH Open document ID: 0824.35068
author: Mishio Kawashita
published in: Osaka Journal of Mathematics
publication date: 1995-11-12T00:00:00Z
review text: The paper is concerned with the rate of local energy decay of solutions to the linear elastic wave equation with the Dirichlet or the Neumann boundary condition in the exterior domain \(\Omega\) in \(\mathbb{R}^ n\) \((n \geq 3)\). More precisely, the equation considered in the paper is the following: \[ \begin{cases} \bigl( \partial^ 2_ t - A (\partial_ x) \bigr) u(t,x) = 0 \quad & \text{in } \mathbb{R} \times \Omega, \\ B (\partial_ x) u(t,x) = 0 \quad & \text{on } \mathbb{R} \times \partial \Omega, \\ u(0,x) = f_ 1(x),\;u_ t(0,x) = f_ 2(x) \quad & \text{in } \Omega. \end{cases} \tag{1} \] Here, \(A (\partial_ x) u = \sum^ n_{i,j = 1} a_{ij} \partial_ ju\), \(a_{ij} = (a_{ipjq})\) are \(n \times n\) constants matrices, \[ B (\partial_ x)u = u |_{\partial \Omega} \text{ (Dirichlet), or } = \sum^ n_{i,j = 1} \nu_ i (x)a_{ij} \partial_ ju |_{\partial \Omega} \text{ (Neumann)}, \] and \(\nu (x) = {^ t(\nu_ 1}(x), \dots, \nu_ n (x))\) is the unit outer normal to \(\partial \Omega\) at \(x \in \partial \Omega\). \(a_{ij}\) satisfy the following assumptions: \(a_{ipjq} = a_{pijq} = a_{jqip}\); \[ \sum^ n_{i,p,j,q = 1} \varepsilon_{jq} \overline {\varepsilon_{ip}} \geq \delta_ 1 \sum^ n_{i,p = 1} | \varepsilon |^ 2; \] \[ A (\xi) = \sum^ n_{i,j = 1} a_{ij} \xi_ i \xi_ j \text{ has } d \text{ characteristic roots of constant multiplicity for any } \xi \in \mathbb{R}^ n - \{0\}, \] where \((\varepsilon_ {ij})\) is any \(n \times n\) symmetric matrix and \(\delta_ 1\) is some positive constant independent of \((\varepsilon_{ij})\). Let \(u(t,x)\) be a solution of (1) with initial data \(f(x) = (f_ 1(x), f_ 2(x))\) and define the solution map \(U(t)\) by the formula: \(U(t)f = (u(t, \cdot), u_ t(t, \cdot))\). Then, the following theorem is the main result. Theorem 1. Assume that the space dimension \(n\) is even and \(\geq 4\). Assume that there exist a function \(p \in C ([0, \infty))\) and a constant \(\widetilde {\rho} > 0\) satisfying the conditions: \[ \lim_{t \to \infty…
reviewed by: Yoshihiro Shibata
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 591441
zbMATH Keywords: rate of local energy decay
zbMATH Keywords: linear elastic wave equation
zbMATH Keywords: exterior domain
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 591441