Nonlinear Analysis 68 (2008) 177–193 www.elsevier.com/locate/na General decay rate estimates for viscoelastic dissipative systems M.M. Cavalcanti a,∗ , V.N. Domingos Cavalcanti a , P. Martinez b a Departamento de Matem ´ atica, Universidade Estadual de Maring´ a, 87020-900 Maring ´ a, PR, Br´ esil b Universite Paul Sabatier, MIP, 118 Route de Narbonne, 31062 Toulouse, France Received 31 July 2006; accepted 30 October 2006 Abstract The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity. The proofs are based on the multiplier method and on a general lemma about convergent and divergent series for obtaining the uniform decay rates. c  2006 Elsevier Ltd. All rights reserved. MSC: 74D10; 74J05; 35B40; 34B15 Keywords: Nonlinear stabilization; Asymptotic behavior at zero and infinity 1. Introduction This paper is concerned with the uniform decay rates of solutions of the viscoelastic problem with nonlinear boundary damping              y tt − Δy +  t 0 h (t − τ)Δy (τ)dτ = 0 in Ω × (0, ∞) y = 0 on Γ 1 × (0, ∞) ∂ y ∂ν −  t 0 h (t − τ) ∂ y ∂ν (τ) dτ + g( y t ) = 0 on Γ 0 × (0, ∞) y (x , 0) = y 0 (x ); y t (x , 0) = y 1 (x ) in Ω , (∗) where Ω is a bounded domain of R n , n ≥ 1, with a smooth boundary Γ = Γ 0 ∪ Γ 1 . Here, Γ 0 and Γ 1 are closed and disjoint and ν represents the unit outward normal to Γ . We assume the following geometrical condition holds true: ∗ Corresponding address: State University of Maringa, Mathematics, Av. Colombo 5790, 87020-900, Maringa Zona 07, Brazil. Tel.: +55 44 32614504; fax: +55 44 32614504. E-mail addresses: cavalcanti273@msn.com, mmcavalcanti@uem.br (M.M. Cavalcanti), vndcavalcanti@uem.br (V.N. Domingos Cavalcanti), martinez@mip.ups-tlse.fr (P. Martinez). 0362-546X/$ - see front matter c  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2006.10.040