Stabilization of the total energy for a system of elasticity with localized dissipation
type: Item
instance of: scholarly article
title: Stabilization of the total energy for a system of elasticity with localized dissipation
zbMATH Open document ID: 1031.35094
author: M. A. Astaburuaga
author: Ruy Coimbra Charão
published in: Differential and Integral Equations
publication date: 2003-05-13T00:00:00Z
review text: The paper studies the decay property for a class of elastic waves in a bounded domain \(\Omega\subset \mathbb{R}^3\). The system of equations with state \(u(x, t)= (u_1,u_2,u_3)\) is described by \[ \begin{gathered} u_{tt}- a^2\Delta u- (b^2- a^2)\nabla\text{ div }u+ q(x) u_t= 0,\\ u(x,0)= f(x)\in [H^2(\Omega)\cap H^1_0(\Omega)]^3,\quad u_t(x, 0)= g(x)\in [H^1_0(\Omega)]^3,\\ u(x, t)= 0,\quad x\in\Gamma= \partial\Omega,\end{gathered} \] where the constants \(a\), \(b\) are such that \(0< a< b\), and \(q\geq 0\) means the dissipation term. For any \(x_0\in\mathbb{R}^3\), set \(\Gamma_1= \{x\in\Gamma: (x- x_0)\cdot\eta(x)> 0\}\), where \(\eta(x)\) denotes the unit outward normal on \(\Gamma\). Let \(\omega\in \overline\Omega\) be a neighborhood of \(\Gamma_1\). The assumption on \(q(x)\) is such that \[ q\in L^r(\Omega),\quad \int_\omega {dx\over q(x)^p}< \infty,\quad\text{and }q(x)= 0\quad\text{in }\Omega\setminus\omega \] for some constants \(p\geq 1\) and \(r\geq {1\over 4}(9+ \sqrt{33})\). Thus the dissipation occurs locally only near the boundary \(\Gamma\). Let \(E(t)\) denote the total energy of the system. Then \(E(t)\) is a nonincreasing function. In fact, it is shown that \(E'(t)= -\int_\omega q(x)|u_t|^2 dx\leq 0\). Based mainly on the multiplier technique, it is then shown that \(E(t)\) decays polynomially as \(t\to\infty\), where the decay rate is determined by \(p\) and \(r\).
Mathematics Subject Classification ID: 35L55
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 93C20
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1839895
zbMATH Keywords: linear hyperbolic systems
zbMATH Keywords: polynomial decay
zbMATH Keywords: multiplier technique
reviewed by: Takao Nambu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1839895