Global Existence and Decay for a System of Two Singular Nonlinear Viscoelastic Equations with General Source and Localized Frictional Damping Terms

Abstract

The current paper deals with the proof of a global solution of a viscoelasticity singular one-dimensional system with localized frictional damping and general source terms, taking into consideration nonlocal boundary condition. Moreover, similar to that in Boulaaras’ recent studies by constructing a Lyapunov functional and use it together with the perturbed energy method in order to prove a general decay result.

Global existence and exponential stability of coupled Lamé system with distributed delay and source term without memory term

“…The functions f 1 (u, v) and f 2 (u, v), which represent the source terms, will be specified later. After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under some given conditions (see, for example, [1][2][3][4][5][6][7][8][9][10][11]). In recent years, several authors have been interested in studying the existence and stability for Lamé systems, we refer to [12][13][14].…”

Polynomial Decay Rate for a Coupled Lamé System with Viscoelastic Damping and Distributed Delay Terms

“…After several authors have studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities and is proven under conditions imposed on the subgroup [1]. The researchers also studied behavior of the energy in a limited field with nonlinear damping and external force and a varying delay of time to find solutions to the Lame system [1][2][3][4][5][6][7][8][9].…”

Exponential decay of solutions for a viscoelastic coupled Lame system with logarithmic source and distributed delay terms

“…Several authors studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities (see previous studies 1‐11 ), and they are proven under conditions imposed on the subgroup 12,13 …”

“…The second integral represent the distributed delay and 2 , 4 ∶ [ 1 , 2 ] → R are bounded functions, where 1 and 2 are two real numbers satisfying 0 ≤ 1 < 2 , and g i , i = 1, 2 are the relaxation functions. Several authors studied the problems of coupled systems and hyperbolic systems, their stability is associated with velocities (see previous studies [1][2][3][4][5][6][7][8][9][10][11], and they are proven under conditions imposed on the subgroup. 12,13 The researchers also studied behavior of the energy in a limited field with nonlinear damping and external force, and a varying delay of time to find solutions to the Lame system.…”

“…The importance of this term appears in many works, and this is due to the fact that many phenomena depends on their past. Also, the distributed delay is influence on the asymptotic behavior of the solution for the different types of mathematics problems such that Timoshenko system, 2,8,18,19 transmission problem, 9 wave equation, 10 and thermoelastic system ( 20,21 ).…”