Shadi Al-Omari | Searcharxiv

arXiv2021

Stability result for a viscoelastic wave equation in the presence of finite and infinite memories

In this paper, we are concerned with the following viscoelastic wave equation \begin{equation*} \label{1} u_{tt}-\nabla u +\int_0^t g_1 (t-s)~ div(a_1(x) \nabla u(s))~ ds + \int_0^{+ \infty} g_2 (s)~ div(a_2(x) \nabla u(t-s)) ~ds = 0, \end{equation*} in a bounded domain $Ω$. Under suitable conditions on $a_1$ and $a_2$ and for a wide class of relaxation functions $g_1$ and $g_2$. We establish a general decay result. The proof is based on the multiplier method and makes use of convex functions and some inequalities. More specifically, we remove the constraint imposed on the boundedness condition on the initial data $\nabla u_{0}$. This study generalizes and improves previous literature outcomes.

math.AP↗

arXiv2021

New Decay Results for a Partially Dissipative Viscoelastic Timoshenko System with Infinite Memory

In this paper, we consider the following dissipative viscoelastic with memory-type Timoshenko system \begin{equation*} \begin{gathered} \begin{cases} ρ_1 ϕ_{tt} - κ( ϕ_{x} + ψ) _x + κ\int_0^\infty g(s) (ϕ_x +ψ)_x(t-s) ~ds =0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ ρ_2 ψ_{tt} - b ψ_{xx} + κ( ϕ_{x} + ψ)-κ\int_0^\infty g(s) (ϕ_x +ψ)(t-s)~ ds=0 & \text{in}~ \left( {0,L} \right) \times \mathbb{R}^+ , \\ \end{cases} \end{gathered} \end{equation*} with Dirichlet boundary conditions, where $g$ is a positive non-increasing function satisfying, for some nonnegative functions $ξ$ and $H$, \[g'(t)\leq-ξ(t)H(g(t)),\qquad\forall~ t\geq0.\] Under appropriate conditions on $ξ$ and $H$, we establish some new decay results for the case of equal-speeds of propagation that generalize and improve many earlier results in the literature.

math.AP↗