DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2016029 DYNAMICAL SYSTEMS SERIES S Volume 9, Number 3, June 2016 pp. 791–813 STABILITY OF THE WAVE EQUATION WITH LOCALIZED KELVIN-VOIGT DAMPING AND BOUNDARY DELAY FEEDBACK Serge Nicaise Universit´ e de Valenciennes et du Hainaut Cambr´ esis LAMAV, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9, France Cristina Pignotti Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica Universit` a di L’Aquila Via Vetoio, Loc. Coppito, 67010 L’Aquila, Italy Abstract. We study the stabilization problem for the wave equation with lo- calized Kelvin–Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an ex- ponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered. 1. Introduction. Let Ω ⊂ IR n be an open bounded set with a boundary Γ of class C 2 . We assume that Γ is divided into two open parts Γ 0 and Γ 1 , i.e. Γ = ¯ Γ 0 ∪ ¯ Γ 1 , with Γ 0 ∩ Γ 1 = ∅ and meas Γ i =0,i =0, 1. In this domain Ω, we consider the initial boundary value problem u tt (x, t) − Δu(x, t) − div (a(x)∇u t (x, t)) = 0 in Ω × (0, +∞), (1.1) u(x, t)=0 on Γ 0 × (0, +∞), (1.2) ∂u ∂ν (x, t)= −a(x) ∂u t ∂ν (x, t) − ku t (x, t − τ ) on Γ 1 × (0, +∞), (1.3) u(x, 0) = u 0 (x) and u t (x, 0) = u 1 (x) in Ω, (1.4) u t (x, t)= f 0 (x, t) in Γ 1 × (−τ, 0), (1.5) where ν (x) denotes the outer unit normal vector to the point x ∈ Γ and ∂u ∂ν is the normal derivative of u. Moreover, τ> 0 is the time delay, k is a real number and a(x) ∈ L ∞ (Ω) satisfies a(x) ≥ 0 a. e. Ω, a(x) ≥ a 0 > 0 a. e. ω, where ω ⊂ Ω is an open neighborhood of the part Γ 1 of the boundary that is supposed to be connected and such that meas (¯ ω ∩ Γ 0 ) > 0. The initial datum (u 0 ,u 1 ,f 0 ) belongs to a suitable space. 2010 Mathematics Subject Classification. Primary: 35L05; Secondary: 93D15. Key words and phrases. Wave equation, delay feedbacks, stabilization. 791