Uniform Stabilization of the Wave Equation by Nonlinear Boundary Feedback

The question of uniformly stabilizing the solution of the wave equation y'' - Δ y = 0 in Ω × (0,∞ ) (Ω is a bounded domain of Rⁿ) by means of a nonlinear feedback law of the following form is studied: ∂ y / ∂ v = - k(x)g(y') on Γ ₀ × (0,∞ ), $y = 0$ on Γ ₁ × (0,∞ ), (Γ ₀ ,Γ ₁ ) being a suitable partition of the boundary of Ω and g a continuous nondecreasing function such that $g(0) = 0$. We choose k(x) ∈ L^∞, k(x) 0 such that $k(x)$ vanishes linearly at the interface points x ∈ Γ ₀ ∩ Γ ₁. Then, if $g(s)$ behaves like | s |p - 1 s as | s | → 0 with $p > 1$ and linearly as | s | → ∞, it is proved that the energy of every solution decays like t^ - 2 / (p - 1) as t → ∞. In the case where $p = 1$ the exponential decay rate is proved.