JournalszaaVol. 30, No. 2pp. 181–191

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

  • Sergio Frigeri

    Università di Pavia, Italy
On the Convergence to Stationary Solutions for a Semilinear Wave Equation with an Acoustic Boundary Condition cover
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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 Ω\Omega, i.e., the velocity potential uu, coupled with an ordinary differential equation for the evolution of an unknown function on Ω\partial\Omega, i.e., the normal displacement δ\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)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.

Cite this article

Sergio Frigeri, On the Convergence to Stationary Solutions for a Semilinear Wave Equation with an Acoustic Boundary Condition. Z. Anal. Anwend. 30 (2011), no. 2, pp. 181–191

DOI 10.4171/ZAA/1430