Abstract
In this work, we consider a quasilinear system of viscoelastic equations with degenerate damping and source terms without the Kirchhoff term. Under suitable hypothesis, we study the blow-up of solutions.
1. Introduction
In this paper, we consider the following problem:where ; for , and for ; and for , and for ; and are positive relaxation functions which will be specified later. is the degenerate damping term, and
The motivation of our problem firstly is by the initial boundary value problem for the quasilinear equation of the form
This 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 ). For example, in the case (), problem (3) has been investigated in [1] and the author proved the blow-up result. In the case () of boundary value problem and in the presence of the dispersion term (), 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 () = 0 instead of ). Furthermore, the authors obtained a blow-up result. In the case (), in [4], Wu studied a general decay of solution. Later, the same author in [5] considered the same problem but () and discussed the decay rate of solution. Recently, in [6], the authors proved the existence of global solution and a general stability result.
There are several works in case (), where the authors have studied the blow-up of solutions of problem (3) (for example, see [3, 7–12]).
For a coupled system, He [13] considered the following problem:where ; and . The author proved general and optimal decay of solutions. Then, 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 , 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 [17–20]). Very recently, in the presence of the dispersion term , 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.
2. Preliminaries
We prove the blow-up result under the following suitable assumptions: (A1) are differentiable and decreasing functions such that (A2) There exist a constants such that
Lemma 1. There exists a function such thatwhere
We take for convenience.
Lemma 2 (see [18]). There exist two positive constants and 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. LetThen, for any initial datum,Problem (1) has a unique solution, for some :where
Now, we define the energy functional.
Lemma 3. Assume (5), (6), and (10) hold; let be a solution of (1); then, is nonincreasing, that is,which satisfies
Proof. By multiplying the first and second equations in (1) by and integrating over , we getWe obtain (14) and (15).
3. Blow-Up
In this section, we prove the blow-up result of solution of problem (1).
First, we define the functional as
Theorem 4. Assume that (5), (6), and (10) hold, and suppose that andThen, the solution of problem (1) blows up in finite time.
Proof. From (14), we haveTherefore,Hence,By (9) and (17), we haveWe setwhere is to be assigned later andBy multiplying the first and second equations in (1) by and with a derivative of (23), we getwhere we haveFrom (25), we find thatAt this point, we use Young’s inequality; for ,We get that for ,Hence, we haveTherefore, using (21) and by setting so thatand substituting in (28), we getwhereWe haveBy Young’s inequality, we find that for ,Hence,Since (10) holds, we obtain the following by using (22) and (24):for some positive constants . By using (24) and the algebraic inequalitywe have, ,where . Also, sincewe conclude thatSubstituting (40) and (42) in (38), we getHence, by fixing , we getfor some constants .
Now, for , from (17),Substituting in (33) and by using (9), we getwhereIn this stage, we take small enough so thatand we assume thatgivesThen, we choose so large such thatFinally, we fix and we appoint small enough so thatThus, for some , estimate (46) becomesBy (9), for some , we obtainandNext, using Hölder’s and Young’s inequalities, we havewhere .
We take , to getSubsequently, by using (24), (22), and (39), we obtainTherefore,Hence, by substituting (59) into (23), we getFrom (53) and (60), we getwhere , and this depends only on and .
By integration of (61), we obtainHence, blows up in timeThen, the proof is completed.
4. Conclusion
The objective of this work is the study of the blow-up of solutions for a quasilinear viscoelastic system with degenerate damping. This 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
The authors declare that they have no conflicts of interest.
Acknowledgments
Sahar Ahmed Idris extends appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant RGP2/53/42.