## 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 $$\Gamma_ 0\neq 0$$.

### 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

### Keywords:

decay; wave equations; boundary dissipation; energy decay
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### References:

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