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Attractors for semilinear damped wave equations with an acoustic boundary condition. (English) Zbl 1239.35025
Summary: 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 \(\delta=\delta (x,t)\), \(x \in \Gamma:= \partial\Omega\), on the boundary. A compatibility condition is also added due to physical reasons. This problem is inspired by a model originally proposed by J. T. Beale and S. I. Rosencrans [Bull. Am. Math. Soc. 80, 1276–1278 (1974; Zbl 0294.35045)]. 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. Finally, we prove the existence of an exponential attractor. The analysis is carried out in dependence of two damping coefficients.

MSC:
35B41 Attractors
35B40 Asymptotic behavior of solutions to PDEs
35L71 Second-order semilinear hyperbolic equations
37L30 Infinite-dimensional dissipative dynamical systems–attractors and their dimensions, Lyapunov exponents
35L20 Initial-boundary value problems for second-order hyperbolic equations
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