Appl. Math. Inf. Sci. 7, No. 4, 1421-1428 (2013) 1421 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/070421 Semilinear hyperbolic boundary value problem for linear elasticity equations A. Rahmoune and B. Benabderrahmane ∗ Laboratoire d’Informatique et de Math´ ematiques, (LIM) Laghouat University(03000), Algeria Received: 17 Nov. 2012, Revised: 26 Feb. 2013, Accepted: 14 Mar. 2013 Published online: 1 Jul. 2013 Abstract: Consider a semilinear hyperbolic boundary value problem associated to the linear elastic equations. Then, existence of a weak solution is established through compactness method. The uniqueness and the regularity of the solution have been gotten by eliminating some hypotheses that have been imposed by other authors for different particular problems. Keywords: Compactness, Existence, Gronwall’s inequality, linear elasticity, Uniqueness of solution, Regularity, Semilinear hyperbolic equation, Variational problem. 1. Introduction In [8], Lions considered a semilinear boundary value problem associated to the Laplace operator with Neumann boundary conditions:    ∂ 2 u ∂t 2 − ∆u + |u| ν u = f inΩ × (0,T ) , u =0 on Γ × (0,T ) , u(x, 0) = u 0 (x),u ′ (x, 0) = u 1 (x),x ∈ Ω. (1) Using the compactness method and Faedo Galerkin techniques, the existence of a weak solution has been proved. Assuming that the condition ν ≤ 2 n − 2 holds, then it follows the uniqueness and the regularity of the solution. In this work, we consider a semilinear hyperbolic boundary value problem governed by partial differential equations that describe the evolution of linear elastic materials with Dirichlet and Neumann boundary conditions as follows :        ∂ 2 u ∂t 2 − divσ (u)+ |u| ν u = f, in Ω × (0,T ) , σ (u)= F (ε(u)) , in Ω × (0,T ) , u = g on Γ 1 × (0,T ) ,σ(u)η =0 on Γ 2 × (0,T ) , u(x, 0) = u 0 (x),u ′ (x, 0) = u 1 (x),x ∈ Ω, (2) where F is a linear function. Assume certain hypotheses on the data functions. Then, by using Faedo Galerkin techniques and compactness method, we will prove the existence of the solution. Our main goal is, without taking into account the condition on ν , to prove the uniqueness and the regularity of the solution. 2. Problem statement Let Ω be an open and bounded domain in R n , recall that the boundary Γ of Ω is assumed to be regular and is composed of two relatively closed parts : Γ 1 , Γ 2 , with mutually disjoint relatively open interiors. We assume that meas (Γ 1 ) > 0. We pose Σ i = Γ i × (0,T ) ,i =1, 2, where T is a finite real. To simplify the writing one will put u ′ = ∂u ∂t ,u ′′ = ∂ 2 u ∂t 2 . σ =(σ ij ), i, j =1, 2, ..., n stands for the stress tensor field. To simplify the notations, we do not indicate explicitly the dependence of the functions u and σ with respect to x ∈ Ω and t ∈ (0,T ). Let η be the unit outward normal vector on Γ . Here and throughout this work, the summation convention over repeated indices is used. The classical formulation of the problem is as follows. Find a displacement field u : Ω × (0,T ) → R n , a stress field σ : Ω × (0,T ) → S n , such that u ′′ − divσ (u)+ |u| ν u = f inQ, ν ∈ N, (3) ∗ Corresponding author e-mail: bbenyattou@yahoo.com c  2013 NSP Natural Sciences Publishing Cor.