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Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary. (English) Zbl 0694.35102
Let $$\Omega \subseteq {\mathbb{R}}^ n$$ be an open bounded domain with smooth boundary $$\Gamma =\Gamma_ 0\cup \Gamma_ 1$$, where $$\Gamma_ 0$$ and $$\Gamma_ 1$$ are disjoint components of the boundary relatively open in $$\Gamma$$. The author considers wave equation with nonlinear dissipative boundary conditions $y_{tt}=\Delta y,\quad x\in \Omega,\quad t>0;\quad y(x,0)=y_ 0(x),\quad y_ t(x,0)=y_ 1(x),\quad x\in \Omega,$
$y(x,t)=0,\quad x\in \Gamma_ 0,\quad t>0;\quad y(x,t)+\gamma [Fy_ t(x,t)]\ni 0,\quad x\in \Gamma_ 1,\quad t>0.$ Here $$\gamma$$ (u) is a monotone increasing, possibly multivalued function defined on $${\mathbb{R}}^ 1$$ such that $$0\in \gamma (0)$$. In the case $$\gamma (0)=0$$, it is shown that a linear dissipative feedback operation F: $$L_ 2(\Omega)\to L_ 2(\Gamma)$$ will “steer” the state $$(y,y_ t)$$ to zero as $$t\to \infty$$. Energy estimates on all limit solutions are obtained for the case $$\gamma$$ (0)$$\neq 0$$. Similar results on plate like equations $$y_{tt}=-\Delta^ 2y$$ are also given.
Reviewer: P.K.Wong

MSC:
 35L05 Wave equation 35L35 Initial-boundary value problems for higher-order hyperbolic equations 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
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