On the Convergence to Stationary Solutions for a Semilinear Wave Equation with an Acoustic Boundary Condition

Abstract

We consider a semilinear wave equation equipped with an acoustic boundary condition. More precisely, we study a system consisting of the wave equation for the evolution of an unknown function in a three-dimensional domain $\Omega$, i.e., the velocity potential $u$, coupled with an ordinary differential equation for the evolution of an unknown function on $\partial \Omega$, i.e., the normal displacement $\delta$. The system is completed with a third condition expressing the impenetrability of the boundary. This problem, inspired on a model for acoustic wave motion of a fluid in a domain with locally reacting boundary surface, originally proposed by J.T. Beale and S.I. Rosencrans in [Bull. Amer. Math. Soc. 80 (1974), 1276–1278], has been studied by S. Frigeri in [J. Evol. Equ. 10 (2010), 29–58] from the point of view of the global asymptotic analysis. The goal of this paper is to analyze the asymptotic behavior of single trajectories, proving that, when the nonlinearity $f(u)$ is analytic, every weak solution converges to a stationary state. The result is obtained by suitably using an argument due to Haraux-Jendoubi and based on the Simon–Lojasiewicz inequality. Furthermore, we provide an estimate for the decay rate to equilibrium.