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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

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
Full Text: DOI
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