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

DOI: 10.2140/pjm.2013.266.257

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