Abstract
This paper is devoted to the study of uniform energy decay rates of solutions to the wave equation with Cauchy–Ventcel boundary conditions:
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 ω 1 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 ω 0 be a neighbourhood of \({\overline{\Gamma_0^{\ast}}}\) . We prove that if a(x) ≥ a 0 > 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)].
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Research of Marcelo M. Cavalcanti was partially supported by the CNPq Grant 300631/2003-0.
Research of Valéria N. Domingos Cavalcanti was partially supported by the CNPq Grant 304895/2003-2.
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Cavalcanti, M.M., Cavalcanti, V.N.D., Fukuoka, R. et al. Stabilization of the damped wave equation with Cauchy–Ventcel boundary conditions. J. Evol. Equ. 9, 143–169 (2009). https://doi.org/10.1007/s00028-009-0002-1
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DOI: https://doi.org/10.1007/s00028-009-0002-1