Weak asymptotic decay for a wave equation with gradient dependent damping

We study the problem of asymptotic behavior as t→+∞ of solutions of the weakly damped wave equation u_tt−Δu=−a(x)β(u_t,∇u) in Ω, u=0 on ∂Ω, where Ω is a bounded, open, connected set in R^N, N≥1, with smooth boundary and where λ=(λ₁,…,λ_N+1)→β(λ₁,…,λ_N+1) satisfies the basic assumption ∀λ, λ₁β(λ)≥0. We prove the weak asymptotic stabilization in H¹₀(Ω)×L²(Ω) of all global solutions under very weak assumptions on β (in so far as we need neither hypothesis of monotonicity on β nor condition restricting its asymptotic growth at infinity). In particular, we generalize an earlier result of M. Slemrod. Moreover the method also applies to other equations and above all to other feedbacks (boundary or pointwise feedbacks) and even to hybrid systems.