Vol. 266, No. 2, 2013

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 320: 1
Vol. 319: 1  2
Vol. 318: 1  2
Vol. 317: 1  2
Vol. 316: 1  2
Vol. 315: 1  2
Vol. 314: 1  2
Vol. 313: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Contacts
 
Submission Guidelines
Submission Form
Policies for Authors
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
 
Other MSP Journals
Rate of attraction for a semilinear wave equation with variable coefficients and critical nonlinearities

Fágner Dias Araruna and Flank David Morais Bezerra

Vol. 266 (2013), No. 2, 257–282
Abstract

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

We study the rate of convergence of global attractors and eigenvalues of the family of dissipative semilinear wave equations with variable coefficients $u_{tt}+\Lambda_\epsilon u+\Lambda_\epsilon^{\delta}u_t=f(u)$, where $\Lambda_\epsilon$ is the elliptic operator $-\operatorname{div} (a_\epsilon(x)\nabla)$ with $\epsilon \in [0,1]$ and sufficiently smooth coefficients $a_\epsilon$, and where $\delta\in\big(\frac12,1\big)$ and the nonlinearity $f$ is a continuously differentiable function satisfying suitable growth conditions. We show that the rate of convergence, as $\epsilon\to0^+$, of the global attractors of these problems, as well as of their eigenvalues, is proportional to the distance of the coefficients $\|a_\epsilon-a_0\|_{L^\infty(\Omega)}$.

Keywords
wave equations, variable coefficients, global attractors, rates of convergence
Mathematical Subject Classification 2010
Primary: 35J05
Secondary: 34D45, 41A25
Milestones
Received: 3 October 2012
Revised: 4 March 2013
Accepted: 13 June 2013
Published: 12 November 2013
Authors
Fágner Dias Araruna
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa, PB 58051-900
Brazil
Flank David Morais Bezerra
Departamento de Matemática
Universidade Federal da Paraíba
João Pessoa, PB 58051-900
Brazil