Differential and Integral Equations Volume 24, Numbers 1-2 (2011), 69–81 GENERAL DECAY TO A VON K ´ ARM ´ AN PLATE SYSTEM WITH MEMORY BOUNDARY CONDITIONS M. L. Santos Faculdade de Matem´ atica-PPGME, Universidade Federal do Par´ a Campus Universit´ario do Guam´ a, Rua Augusto Corrˆ ea 01 Cep 66075-110, Par´ a, Brazil A. Soufyane Faculty of Engineering and Applied Sciences, Alhosn University P.O.Box 38772, Abu Dhabi, United Arab Emirates (Submitted by: Tohru Ozawa) Abstract. In this paper we consider the von K´arm´ an plate system subject to boundary conditions of memory type. We establish a general decay result, from which the usual exponential and polynomial decay rates are only special cases. Our work allows certain relaxation func- tions which are not necessarily of exponential or polynomial decay and therefore generalizes and improves earlier results in the literature. 1. Introduction The main purpose of this work is to study the general decay of the solu- tions to a von K´arm´an plate system with boundary conditions of memory type. To introduce this model we need some notation. Let Ω be an open bounded subset of R 2 with regular boundary Γ. Let us denote by ν =(ν 1 ,ν 2 ) the external unit normal vector on Γ and by τ =(−ν 2 ,ν 1 ) the correspond- ing unit tangent vector. Taking into account this notation, we consider the following initial-boundary-value problem: u tt +Δ 2 u =[u, v] in Ω × (0, ∞), (1.1) Δ 2 v = −[u, u] in Ω × (0, ∞), (1.2) v = ∂v ∂ν =0 on Γ × (0, ∞), (1.3) ∂u ∂ν +  t 0 g 1 (t − s)  B 1 u(s)+ ρ 1 ∂u ∂ν (s)  ds =0 on Γ × (0, ∞), (1.4) Accepted for publication: March 2010. 69