#### Vol. 266, No. 2, 2013

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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