Blow-Up of Solutions for a Coupled Nonlinear Viscoelastic Equation with Degenerate Damping Terms: Without Kirchhoff Term

Department of Mathematics, College of Sciences and Arts, ArRas, Qassim University, Buraydah, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran 31000, Algeria Department of Mathematics, Faculty of Exact Sciences, University of El Oued, El Oued, Algeria Department of Mathematics, Faculty of Science, King Khalid University, Abha 61471, Saudi Arabia Mathematics Department, Faculty of Science, South Valley University, Qena 83523, Egypt Center for Nonlinear Systems, Chennai Institute of Technology, Chennai, Tamil Nadu 600069, India College of Industrial Engineering, King Khalid University, Abha 62529, Saudi Arabia Department of Mathematics, College of Sciences, Juba University, Juba, Sudan

(3) is type of problem is frequently found in some mathematical models in applied sciences, especially in the theory of viscoelasticity. Problem (3) has been studied by various authors, and several results concerning asymptotic behavior and blow-up have been studied (case η ≥ 0). For example, in the case (g(u, u t ) � 0), problem (3) has been investigated in [1] and the author proved the blow-up result. In the case (g(u, u t ) � 0) of boundary value problem and in the presence of the dispersion term (− Δu tt ), Liu [2] studied a general decay of solutions. And, in [3], the authors applied the potential well method to indicate the global existence and uniform decay of solutions (g(u; u t )) � 0 instead of Δu t ). Furthermore, the authors obtained a blow-up result. In the case (g(u, u t ) � |u t | m u t ), in [4], Wu studied a general decay of solution. Later, the same author in [5] considered the same problem but (g(u, u t ) � u t ) and discussed the decay rate of solution. Recently, in [6], the authors proved the existence of global solution and a general stability result.
ere are several works in case (η � 0), where the authors have studied the blow-up of solutions of problem (3) (for example, see [3,[7][8][9][10][11][12]). For a coupled system, He [13] considered the following problem: where η > 0; j, s ≥ 2; and e author proved general and optimal decay of solutions. en, in [14], the author investigated the same problem without damping term and established a general decay of solutions. Furthermore, the author obtained a blow-up of solutions. In addition, in problem (1) with η � 0, in [15], Wu proved a general decay of solutions. Later, in [16], Piskin and Ekinci established a general decay and blow-up of solutions with nonpositive initial energy for problem (1) case (Kirchhoff type). In recent years, some other authors investigate the hyperbolic type system with degenerate damping terms (see [17][18][19][20]). Very recently, in the presence of the dispersion term (− Δu tt ), our problem (1) has been studied in [21]. Under some restrictions on the initial datum and standard conditions on relaxation functions, the authors have established the global existence and proved the general decay of solutions.
Based on all of the abovementioned discussion, we believe that the combination of these terms of damping (memory term, degenerate damping, and source terms) constitutes a new problem worthy of study and research, different from the above that we will try to shed light on, especially the blow-up of solutions.
Our paper is divided into several sections: In Section 2, we lay down the hypotheses, concepts, and lemmas we need.
In Section 3, we prove our main result. Finally, we give some concluding remarks in the last section.

Preliminaries
We prove the blow-up result under the following suitable assumptions: (A1) h i : R + ⟶ R + are differentiable and decreasing functions such that (A2) ere exist a constants ξ 1 , ξ 2 > 0 such that where We take a 1 � b 1 � 1 for convenience.
Lemma 2 (see [18]). ere exist two positive constants c 0 and c 1 such that Now, we state the local existence theorem that can be established by combining arguments of [13,16].

Theorem 1. Assume (5) and (6) hold. Let
en, for any initial datum, Problem (1) has a unique solution, for some T > 0: 2 Complexity where Now, we define the energy functional.

Lemma 3.
Assume (5), (6), and (10) hold; let (u, v) be a solution of (1); then, E(t) is nonincreasing, that is, which satisfies Proof. By multiplying the first and second equations in (1) by u t , v t and integrating over Ω, we get d dt We obtain (14) and (15).

Blow-Up
In this section, we prove the blow-up result of solution of problem (1).
First, we define the functional as Complexity Theorem 4. Assume that (5), (6), and (10) hold, and suppose that E(0) < 0 and en, the solution of problem (1) blows up in finite time.

Conclusion
e objective of this work is the study of the blow-up of solutions for a quasilinear viscoelastic system with degenerate damping. is type of problem is frequently found in some mathematical models in applied sciences, especially in the theory of viscoelasticity. What interests us in this current work is the combination of these terms of damping (memory term, degenerate damping, and source terms), which dictates the emergence of these terms in the system.
In the next work, we will try using the same method with the same problem in addition to other damping terms (dispersion term, Balakrishnan-Taylor damping, and delay term).

Data Availability
No data were used.

Conflicts of Interest
e authors declare that they have no conflicts of interest.