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Attractors for semilinear damped wave equations with an acoustic boundary condition

Attractors for semilinear damped wave equations with an acoustic boundary condition In this paper, we study a semilinear weakly damped wave equation equipped with an acoustic boundary condition. The problem can be considered as a system consisting of the wave equation describing the evolution of an unknown function u = u ( x , t ), $${x\in\Omega}$$ in the domain coupled with an ordinary differential equation for an unknown function δ = δ ( x , t ), $${x\in\Gamma:=\partial\Omega}$$ on the boundary. A compatibility condition is also added due to physical reasons. This problem is inspired on a model originally proposed by Beale and Rosencrans (Bull Am Math Soc 80:1276–1278, 1974). The goal of the paper is to analyze the global asymptotic behavior of the solutions. We prove the existence of an absorbing set and of the global attractor in the energy phase space. Furthermore, the regularity properties of the global attractor are investigated. This is a difficult issue since standard techniques based on the use of fractional operators cannot be exploited. We finally prove the existence of an exponential attractor. The analysis is carried out in dependence of two damping coefficients. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

Attractors for semilinear damped wave equations with an acoustic boundary condition

Journal of Evolution Equations , Volume 10 (1) – Mar 1, 2010

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References (37)

Publisher
Springer Journals
Copyright
Copyright © 2010 by Birkhäuser / Springer Basel AG
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-009-0039-1
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study a semilinear weakly damped wave equation equipped with an acoustic boundary condition. The problem can be considered as a system consisting of the wave equation describing the evolution of an unknown function u = u ( x , t ), $${x\in\Omega}$$ in the domain coupled with an ordinary differential equation for an unknown function δ = δ ( x , t ), $${x\in\Gamma:=\partial\Omega}$$ on the boundary. A compatibility condition is also added due to physical reasons. This problem is inspired on a model originally proposed by Beale and Rosencrans (Bull Am Math Soc 80:1276–1278, 1974). The goal of the paper is to analyze the global asymptotic behavior of the solutions. We prove the existence of an absorbing set and of the global attractor in the energy phase space. Furthermore, the regularity properties of the global attractor are investigated. This is a difficult issue since standard techniques based on the use of fractional operators cannot be exploited. We finally prove the existence of an exponential attractor. The analysis is carried out in dependence of two damping coefficients.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Mar 1, 2010

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