Attractors for semilinear damped wave equations with an acoustic boundary condition

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.