QUARTERLY OF APPLIED MATHEMATICS VOLUME LXIII, NUMBER 3 SEPTEMBER 2005, PAGES 429–453 S 0033-569X(05)00967-4 Article electronically published on July 11, 2005 STABILIZATION OF A SYSTEM OF ANISOTROPIC THERMOELASTICITY BY NONLINEAR BOUNDARY AND INTERNAL FEEDBACKS By AMAR HEMINNA (U.S.T.H.B., Fac de Maths, El-Alia, Bab Ezzouar, Alger Alg´ erie ), SERGE NICAISE (Universit´ e de Valenciennes et du Hainaut-Cambr´ esis, MACS, Institut des Sciences et Techniques de Valenciennes, F-59313 Valenciennes Cedex 9, France ), and ABDOULAYE S ` ENE (Universit´ e Cheikh Anta DIOP, D´ epartement de Math´ ematiques, Facult´ e des sciences et techniques, Dakar S´ en´ egal ) Abstract. We consider the stabilization of an anisotropic thermoelasticity system with a natural Neumann boundary condition on part of the boundary and combined nonlinear internal and boundary feedbacks. We then give an answer to a problem raised by Liu and Zuazua. 1. Introduction. Let Ω be a nonempty bounded open subset of R n ,n ≥ 1, with a boundary Γ of class C 2 . We denote by ν =(ν 1 , ··· ,ν n ) ⊤ the unit outward normal vector along Γ. For a fixed x 0 ∈ R n we define the function m(x)= x − x 0 ; x ∈ R n and the following partition of the boundary Γ (see Figure 1): Γ 1 = {x ∈ Γ: m(x) · ν (x) ≤ 0}, (1) Γ 2 = {x ∈ Γ: m(x) · ν (x) > 0}. (2) In this paper we consider the system of anisotropic thermoelasticity: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u ′′ − div σ(u)+ α∇θ + f (u ′ ) = 0 in Q := Ω × R + , θ ′ − ∆θ + βdiv u ′ = 0 in Q, θ = 0 on Σ = Γ × R + , u = 0 on Σ 1 =Γ 1 × R + , σ(u) · ν + au + g(u ′ ) = 0 on Σ 2 =Γ 2 × R + , u(·, 0) = u 0 ,u ′ (·, 0) = u 1 θ(·, 0) = θ 0 in Ω, (3) Received February 1, 2004. 2000 Mathematics Subject Classification. Primary 35B35, 73B30. E-mail address : ahemina@hotmail.com E-mail address : s.nicaise@univ-valenciennes.fr E-mail address : abdousen@ucad.sn c 2005 Brown University 429 License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf