Uniform Stabilization of the Wave Equation by Nonlinear Boundary Feedback

Abstract

The question of uniformly stabilizing the solution of the wave equation $y'' - \Delta y = 0$ in $\Omega \times (0,\infty )$ ($\Omega $ is a bounded domain of ${\bf R}^n $) by means of a nonlinear feedback law of the following form is studied: ${{\partial y} / {\partial v = - k(x)g(y')}}$ on $\Gamma _0 \times (0,\infty )$, $y = 0$ on $\Gamma _1 \times (0,\infty )$, $(\Gamma _0 ,\Gamma _1 )$ being a suitable partition of the boundary of $\Omega $ and g a continuous nondecreasing function such that $g(0) = 0$. We choose $k(x) \in L^\infty $, $k(x) \geqq 0$ such that $k(x)$ vanishes linearly at the interface points $x \in \bar \Gamma _0 \cap \bar \Gamma _1 $. Then, if $g(s)$ behaves like $| s |^{p - 1} s$ as $| s | \to 0$ with $p > 1$ and linearly as $| s | \to \infty $, it is proved that the energy of every solution decays like $t^{ - {2 / {(p - 1)}}} $ as $t \to \infty $. In the case where $p = 1$ the exponential decay rate is proved.