UNIFORM STABILIZATION OF THE WAVE-EQUATION BY NONLINEAR BOUNDARY FEEDBACK

The question of uniformly stabilizing the solution of the wave equation y″ - Δy = 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 Γ0 × (0, ∞), y = 0 on Γ1 × (0, ∞), (Γ0, Γ1) being a suitable partition of the boundary of Ω and g a continuous nondecreasing function such that g(0) = 0. We choose k(x) member of L∞ (Γ0), k(x) ≥ 0 such that k(x) vanishes linearly at the interface points x member of Γ̄0 intersection Γ̄1. Then, if g(s) behaves like |s|p-1s 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 were p = 1 the exponential decay rate is proved.