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Decay of solutions of wave equations in a bounded region with boundary dissipation. (English) Zbl 0536.35043

It has been proved an energy decay rate for solution of the wave equation \((1)\quad \partial^ 2w/\partial t^ 2-\Delta_ nw=0\) in \((0,\infty)\times \Omega\) with boundary conditions \((2)\quad w(x,t)=0\) on \(\Gamma_ 0\times [0,\infty)\), \((3)\quad \partial w/\partial \nu +a(x)\partial w/\partial t=0\) on \([0,\infty)\times \Gamma_ 1\). \(\Omega\) is a bounded, open, connected set in \(R^ n(n\geq 2)\) having a boundary \(\Gamma\) which is of class \(C^ 2\) and which consists of two parts \(\Gamma_ 0\) and \(\Gamma_ 1\), with \(\Gamma_ 1\neq 0\) and relatively open in \(\Gamma\). \(\Gamma_ 0\) is assumed to be either empty or to have a nonempty interior. \(\Gamma_ 0\) is a reflecting surface and \(\Gamma_ 1\) an energy absorbing surface, \(\nu\) is the unit normal of \(\Gamma\) pointing towards the exterior of \(\Omega\), \(a\in C^ 1({\bar \Gamma}_ 1)\), with \(a(x)\geq a_ 0>0\) on \(\Gamma_ 1\). The main result:
Theorem 1. Assume there is a vector field \(l(x)=(l_ 1(x),...,l_ n(x))\) of class \(C^ 2({\bar \Omega})\) such that (i) \(l,\nu \leq 0\) a.e. on \(\Gamma_ 0\), (ii) \(l\cdot \nu \geq \gamma>0\) a.e. on \(\Gamma_ 1\), (iii) (\(\partial l_ i/\partial x_ j+\partial l_ j/\partial x_ i)\) is uniformly positive definite on \({\bar \Omega}\). Then there are positive constants C,\(\delta\), such that \(E(w,t)\leq Ce^{-\delta t}E(w,0),\) \(t\geq 0\) for every solution of (1), (2), (3) for which \(E(w,0)<\infty.\) The condition \(E(w,0)<\infty\) means that \(W(\cdot,0)\in H^ 1(\Omega)\), \(w_ t(\cdot,0)\in L^ 2(\Omega)\) and \(w(x,0)=0\), on \(\Gamma_ 0\) if \(\Gamma_ 0\neq 0\).
Reviewer: W.Sadkowski

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35L05 Wave equation
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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