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arXiv · 2109.05811

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

Shadi Al-Omari
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Abstract

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.

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Shadi Al-Omari. 2021-09-13. New Decay Results for a Partially Dissipative Viscoelastic Timoshenko System with Infinite Memory. https://arxiv.org/abs/2109.05811

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