Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions

Abstract

This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions: $$\left\{\begin{array}{l@{\qquad}l@{\quad}l} u_{tt}- \Delta u +a(x)g(u_t)=0 \quad {\rm in}\quad \Omega\;\times ] 0,\infty[\\ \partial_{\nu} u - \Delta_{\Gamma_1}u=0 \qquad \qquad \quad {\rm on}\quad \Gamma_{1}\times ] 0,\infty[\\ u=0\qquad \qquad \qquad \quad \qquad {\rm on} \quad\Gamma_{0}\times ] 0,\infty [\end{array}\right.$$ where Ω is a bounded domain of \({\mathbb{R}^{n}}\) (n ≥ 2) having a smooth boundary \({\Gamma :=\partial \Omega}\) , such that \({ \Gamma =\Gamma _{0} \cup \Gamma_{1}}\) with \({\Gamma_0}\) , \({\Gamma_1}\) being closed and disjoint. It is known that if a(x) = 0 then the uniform exponential stability never holds even if a linear frictional feedback is applied to the entire boundary of the domain [see, for instance, Hemmina (ESAIM, Control Optim Calc Var 5:591–622, 2000, Thm. 3.1)]. Let \({f:\overline{\Omega} \rightarrow \mathbb R}\) be a smooth function; define ω1 to be a neighbourhood of \({\Gamma_1}\) , and subdivide the boundary \({\Gamma_0}\) into two parts: \({\Gamma_0^{\ast}=\{x\in \Gamma_0;\partial_{\nu}f > 0\}}\) and \({\Gamma_0 \backslash \Gamma_0^{\ast}}\) . Now, let ω0 be a neighbourhood of \({\overline{\Gamma_0^{\ast}}}\) . We prove that if a(x) ≥ a0 > 0 on the open subset \({\omega =\omega_0 \cup \omega_1}\) and if g is a monotone increasing function satisfying k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, then the energy of the system decays uniformly at the rate quantified by the solution to a certain nonlinear ODE dependent on the damping [as in Lasiecka and Tataru (Differ Integral Equ 6:507–533, 1993)].