On the convergence to stationary solutions for a semilinear wave equation with an acoustic boundary condition (Q537680)

The paper deals with solutions of the equation \[ u_{tt}+\omega u_{t}-\Delta u+u+f(u)=0 \text{ in }\Omega \times (0,+\infty ) \] with the boundary conditions \[ \delta _{tt}+\nu \delta _{t}+\delta =-u_{t}, \;\delta _{t}=\frac{\partial u}{\partial n}\text{ on }\partial \Omega \times (0,+\infty ). \] The problem is inspired by a model for acoustic wave motion of a fluid in a domain with locally reacting boundary surface, originally proposed by \textit{J.T. Beale} and \textit{S. I. Rosencrans} in [Bull. Am. Math. Soc. 80, 1276--1278 (1974; Zbl 0294.35045)]. Under some restrictions on \(f(u)\) the author proves that there exists a function \(u_{\infty }\) of the set of equilibria such that the solution \(u\) of the above problem starting from the given initial data converges to \(u_{\infty }\) as \(t\rightarrow +\infty .\) The result is obtained by use of Haraux's and Jendoubi's argument and the Simon-Łojasiewicz inequality.