New Decay Results for a Partially Dissipative Viscoelastic Timoshenko System with Infinite Memory
arXiv:2109.05811
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 is a positive non-increasing function satisfying, for some nonnegative functions and , \[g'(t)\leq-ξ(t)H(g(t)),\qquad\forall~ t\geq0.\] Under appropriate conditions on and , we establish some new decay results for the case of equal-speeds of propagation that generalize and improve many earlier results in the literature.
arXiv comment20 pages