#### Abstract

This work deals with the well posedness (existence, uniqueness and continuous dependence of the solution on the input data) for a non- homogeneous Timoshenko system with a

linear frictional damping and a memory type damping. The system is composed of two coupled

hyperbolic equations complemented by initial and nonlocal boundary conditions. The proofs

of the results are mainly based on some energy and a priori estimates and on some density

arguments. More precisely, in the bounded domain QT = (0, L) × (0, T), we consider the initial

boundary value problem for a non-homogeneous damped Timoshenko

where ρ1, ρ2, κ1 and κ2 are positive constants and f , g , ϕ , ψ, F, and G are given functions, and h : R+ → R+ is a twice differentiable function such that

κ2 − Z T 0 h(t)dt = l > 0, h0 (t) < 0, ∀t ≥ 0.

This system of coupled hyperbolic equations represents a Timoshenko model for a thick beam of length L, where u is the transverse displacement of the beam and v is the rotation angle of the filament of the beam. The coefficients ρ1, ρ2, κ1 and κ2 are respectively the density, the polar moment of inertia of a cross section, the shear modulus and the Young’s modulus of elasticity.

The integral conditions represent the averages of the total transverse displacement of the beam and the rotation angle of the filament of the beam. Some results on the well posedness of the posed problem are obtained when two classical conditions are replaced by nonlocal conditions.

Joint work with and Faten Aldosari.

#### Brief Biography

Said Mesloub is a Professor of Mathematics at King Saud University Department of Mathematics (KSA). He received his Master degree from Indiana University at Bloomington (USA) in 1986, and received his PhD (doctorat d'État en Mathématiques) in Mathematics from Constantine University. Professor Said published more than 74 research papers.