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. In this paper, we investigate the inuence of boundary dissi-pations on decay property of the solutions for a semilinear wave equationwith damping and memory condition on the boundary using the multi-plier technique. 1. IntroductionOur goal in this paper is to study the general decay rates of the solution fora semilinear wave equation with boundary damping and memory conditions ofthe form:(1.1) u ′′ −∆u+h(∇u) = 0 in Ω×(0,∞),(1.2) u = 0 on Γ 1 ×(0,∞),∂u∂ν+Z t0 (1.3) k(t −s,x)u ′ (s)ds +g(u ′ ) = 0 on Γ 0 ×(0,∞),(1.4) u(0) = u 0 , u ′ (0) = u 1 in Ω,where Ω is a bounded domain in R n (n ≥ 1) with a suciently smooth boundaryΓ = Γ 0 ∪Γ 1 . HereΓ 0 and Γ 1 areclosedanddisjoint, ν is the unit outwardnormalto Γ, u ′ = ∂u∂t , ∆u =P ni=1∂ 2 u∂x i . h, k and g are given functions.For the last several decades, the boundary stabilization of classical waveequations has generated much interest and consideration in the literature. In-deed, when h = 0 and k(t,x) = 0, the problem has been treated many authors[8, 14, 15] for linear cases and [7, 9, 16] for nonlinear ones. Aassila et al. [1]studied problem (1.1)-(1.4), with h = 0 and a kernel k of exponential decay,and established some decay results. The stability of systems with h 6= 0 is

GENERAL DECAY FOR A SEMILINEAR WAVE EQUATION WITH BOUNDARY FRICTIONAL AND MEMORY CONDITIONS