Weak asymptotic decay for a wave equation with gradient dependent damping

Article type: Research Article

Authors: Vancostenoble, J.

Affiliations: MIP, Université Paul Sabatier, Toulouse III, 118 route de Narbonne, 31 062 Toulouse cedex 4, France E‐mail: [email protected]‐tlse.fr

Abstract: We study the problem of asymptotic behavior as t→+∞ of solutions of the weakly damped wave equation utt−Δu=−a(x)β(ut,∇u) in Ω, u=0 on ∂Ω, where Ω is a bounded, open, connected set in RN, N≥1, with smooth boundary and where λ=(λ1,…,λN+1)→β(λ1,…,λN+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.

Keywords: wave equation, distributed control, nonlinear nonmonotone feedback, weak asymptotic stabilization

Journal: Asymptotic Analysis, vol. 26, no. 1, pp. 1-20, 2001