Convergence to equilibrium for the damped semilinear wave equation with critical exponent and dissipative boundary condition

Abstract

This paper is concerned with the asymptotic behavior of the solution to the following damped semilinear wave equation with critical exponent: \begin{equation} u_{tt} + u_t -\Delta u + f(x,u) = 0, \qquad (x,t) \in \Omega \times \mathbb {R}^+ \end{equation} subject to the dissipative boundary condition \begin{equation} \partial _\nu u+ u + u_t = 0, \qquad t > 0, \ x \in \Gamma \end{equation} and the initial conditions \begin{equation} u|_{t=0} = u_0(x),\quad u_t|_{t=0}=u_1(x), \qquad x \in \Omega , \end{equation} where $\Omega$ is a bounded domain in $\mathbb {R}^3$ with smooth boundary $\Gamma$ , $\nu$ is the outward normal direction to the boundary, and $f$ is analytic in $u$. In this paper convergence of the solution to an equilibrium as time goes to infinity is proved. While these types of results are known for the damped semilinear wave equation with interior dissipation and Dirichlet boundary condition, this is, to our knowledge, the first result with dissipative boundary condition and critical growth exponent.