General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping

Abdelbaki Choucha: Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria; Department of Matter Sciences, College of Sciences, Amar Teleji Laghouat University, Laghouat, Algeria, e-mail: abdelbaki.choucha@gmail.com  * Corresponding author: Salah Boulaaras, Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Al-Qassim, Kingdom of Saudi Arabia; Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran, Ahmed Benbella, Algeria, e-mail: S.Boularas@qu.edu.sa

In [8], Balakrishnan and Taylor proposed a new model of damping and called it the Balakrishnan-Taylor damping, as it relates to the span problem and the plate equation. For more details, the readers can refer to some papers that focused on the study of this damping [9][10][11][12][13].
On the other hand, the stability issue of systems with delay is of theoretical and practical great importance, whereas, the dynamic systems with delay terms have become a major research subject in differential equation since the last five decades. Recently, the stability and the asymptotic behavior of evolution systems with time delay especially the distributed delay effect have been studied by many authors, see [14][15][16][17][18].
Very recently, in [19] the authors considered our problem (1.1) but in the presence of the delay, they proved the general decay result of solutions by the energy method under suitable assumptions.
Based on all of the above, the combination of these terms of damping (memory term, Balakrishnan-Taylor damping, and the distributed delay ) in one particular problem with the addition of α t ( ) to the term of memory and the distributed delay term β s u t s u t s s d we believe that it constitutes a new problem worthy of study and research, different from the above that we will try to shed light on. Our paper is divided into several sections: in Section 2 we lay down the hypotheses, concepts, and lemmas we need, and in Section 3 we prove our main result. Finally, we give a conclusion in Section 4.

Preliminaries
For studying our problem, in this section we need some materials.
So, problem (1.1) can be written as Now, we give the energy functional.
Proof. Taking the inner product of 2.5 1 ( ) with u t , then integrating over Ω, we find By computation, integration by parts and the last condition in (2.5), we get by integration by parts, we find and by inserting (2.14) and (2.15) into (2.13), we find Now, multiplying the equation 2.

General decay
In this section, we state and prove the asymptotic behavior of system (2.5). For this goal, we set and t s e β s yx ρ s t s ρ Θ , , , d d . Proof. A differentiation of (3.1) and using 2.  We estimate the last three terms of the right-hand side (RHS) of (3.5 and Similar to I 1 , we have  Proof. A differentiation of (3.2) and using 2.