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- Publisher
- Springer Journals
- Copyright
- Copyright © 2009 by Birkhäuser Verlag Basel/Switzerland
- Subject
- Mathematics; Analysis
- ISSN
- 1424-3199
- eISSN
- 1424-3202
- DOI
- 10.1007/s00028-009-0002-1
- Publisher site
- See Article on Publisher Site
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 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)).
Journal
Journal of Evolution Equations – Springer Journals
Published: May 1, 2009