Large time behavior of anisotropic electromagnetic/elasticity equations in exterior domains
type: Item
instance of: scholarly article
title: Large time behavior of anisotropic electromagnetic/elasticity equations in exterior domains
zbMATH Open document ID: 1178.35066
author: G. Perla Menzala
author: Cleverson Roberto Da Luz
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-08-20T00:00:00Z
review text: The authors consider the coupled system of anistotropic electromagnetic elasticity model of the form: \[ \begin{aligned} u_{tt}- \sum_{i,j=1}^3 {\partial\over\partial x_i} \left(A_{ij}(x){\partial u\over\partial x_j}\right)+ u_t+\gamma \operatorname{curl} E=0, \tag{1}\\ \varepsilon(x)E_t- \operatorname{curl} H+\sigma E- \gamma \operatorname{curl} u_t=0, \tag{2}\\ \mu(x)H_t+ \operatorname{curl} E=0, \tag{3}\\ \operatorname{div}(\mu(x)H)=0 \tag{4}\end{aligned} \] in \(\Omega\times(0,\infty)\), \(\Omega\subset\mathbb R^3\) -- is an extension (unbounded with bounded complement and with compact smooth boundary \(\partial\Omega\)), with initial conditions \[ (u,u_t,E,H)|_{t=0}=(u_0,u_1,E_0,H_0)\quad \text{in }\Omega \tag{5} \] and boundary conditions \[ u=0,\quad \eta\times E=0\quad \text{on }\partial\Omega\times(0,\infty) \] where: \(\eta=\eta(x)\) denotes unit normal vector at \(x\in\partial\Omega\), \(u=(u_1,u_2,u_3)\) -- the displacement vector, \(E=(E_1,E_2,E_3)\) the dielectric induction, \(H=(H_1,H_2,H_3)\) -- the magnetic induction, \(\epsilon(x)\), \(\mu(x)\) -- denote the electric permittivity and magnetic permeability \(\sigma>0\), \(\gamma>0\) -- are the coefficients of the electric conductivity and coupling constant \(\eta=\eta(x)\) -- denotes the unit vector at \(x\in\partial\Omega\). The total energy associated with system (1) is given by \[ {\mathcal L}(t)={1\over2} \intop_\Omega\bigg[|u_t(t)|^2+\sum_{i,j}^3A_{ij}(x) {\partial u\over\partial x_j}(t)\cdot{\partial u\over\partial x_i}(t)+ \varepsilon(x)E(t)E(t)+\mu(x)H(t)\cdot H(t)\bigg]\,dx. \] The main result of this paper is laying in the fact that they proved uniform rates of decay of \({\mathcal L}(t)\) as \(t\to\infty\). In the main theorem, the authors prove that \({\mathcal L}(t)\) decays uniformy at the rate \((1+t)^{-1}\). They show that if they choose more regular data, several terms in higher order energy do decay at faster rates. In the proof they use the method of energy estimate.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35Q61
Mathematics Subject Classification ID: 74F15
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 5597523
zbMATH Keywords: electromagnetic elasticity model
zbMATH Keywords: coupled system
reviewed by: Jerzy August Gawinecki
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.05.060
OpenAlex ID: W2011854214
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DOI: 10.1016/J.JMAA.2009.05.060
description: scientific article; zbMATH DE number 5597523