Advances in Differential Equations Volume 8, Number 7, July 2003, Pages 873–896 STABILITY FOR A SYSTEM OF WAVE EQUATIONS OF KIRCHHOFF WITH COUPLED NONLINEAR AND BOUNDARY CONDITIONS OF MEMORY TYPE M. L. Santos Departamento de Matem´atica, Universidade Federal do Par´ a Campus Universitario do Guam´ a Rua Augusto Corrˆ ea 01, CEP 66075-110, Par´ a, Brazil J. Ferreira Departamento de Matem´atica-DMA, Universidade Estadual de Maring´ a-UEM Av. Colombo, 5790-Zona 7, CEP 87020-900, Maring´ a-Pr., Brazil (Submitted by: Y. Giga) Abstract. In this paper, we consider a system of two wave equations of Kirchhoff with coupled nonlinear and memory conditions at the bound- ary, and we study the asymptotic behavior of the corresponding solu- tions. We prove that the energy decays with the same rate of decay of the relaxation functions; that is, the energy decays exponentially when the relaxation functions decay exponentially and polynomially when the relaxation functions decay polynomially. 1. Introduction The main purpose of this work is to study the asymptotic behavior of the solutions of a system of two nonlinear wave equations of Kirchhoff type with coupled nonlinear and boundary conditions of memory type. To formalize this problem let us take Ω an open, bounded set of R n with smooth boundary Γ, and let us assume that Γ can be divided in two nonempty parts Γ = Γ 0 ∪Γ 1 with ¯ Γ 0 ∩ ¯ Γ 1 = ∅. Let us denote by ν (x) the unit normal vector at x ∈ Γ outside of Ω, and let us consider the following initial boundary value problem: u tt − M (||∇u|| 2 2 + ||∇v|| 2 2 )Δu − Δu t + f (u − v)=0 in Ω × (0, ∞), (1.1) v tt − M (||∇u|| 2 2 + ||∇v|| 2 2 )Δv − Δv t − f (u − v)=0 in Ω × (0, ∞), (1.2) u = v =0 on Γ 0 × (0, ∞), (1.3) Accepted for publication: March 2003. AMS Subject Classifications: 34A12, 34A34. 873