Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

Abstract

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.

1. Introduction

In this paper, we are concerned with the initial boundary value problem (IBVP) for a quasilinear viscoelastic wave equation whose energy is inhibited by a nonlinear frictional damping and by a Kelvin-Voigt damping. More precisely, we consider IBVPwhere is the unknown, , having the boundary , is a nonempty bounded open subset of with a positive integer, , , are constants, and , the so-called relaxation function or viscoelastic kernel, is a decreasing function. The restrictions on , , and would be added when necessary. Eq. (1)₁ was used to describe the vibration of beams or plates, whose wave propagation speed depends on the velocity of vibration, which is subject to nonlinear frictional damping force and viscoelastic damping force; see [1–3]. In particular, the term with , by Boltzmann’s superposition principle, indicates that the wave speed depends on the velocity of the wave; , the so-called viscoelastic damping, is incorporated in IBVP (1) to characterize the hereditary properties.

In recent years, the wave equation has been frequently chosen as the model to study the competition between different damping terms and the source term . In [4], Messaoudi considered the general decay problem for IBVPand he proved, under certain additional restrictions on the relaxation function , that the energy of IBVP (2) decreases to zero. The results and methods in [4] were later extended by Messaoudi and Al-Khulaifi [5] to IBVPmore precisely, they established some general decay results for IBVP (3). It was shown in both [4, 5] that the energy decay rate is closely related to the peculiarity of the relaxation function . Different from (2)₁, the wave speed of (3)₁ depends on the velocity of the wave. The study of [5] was highly inspired by [6], in which Cavalcanti, Domingos Cavalcanti, and Ferreira studied the IBVPand they proved that the solutions of IBVP (4) exist globally if and their energy decreases to zero if . Amongst the afore-mentioned references, there are vast literatures concerning energy decay for viscoelastic wave equations; see [7–14] and the references cited therein. The key ingredient in proving energy decay estimates is an appropriate modified energy functional associated with the initial boundary value problem under consideration. The usual idea to obtain decay for the afore-chosen energy functional is to exploit various perturbed energy functionals for the above energy functional. This idea is important in our later development.

In the afore-mentioned references, the source term , in one form or another, is “beaten” by the damping terms , and/or . In some cases, these damping terms were also proved to be “beaten” by the above pure power source term; see [15–25]. As with the procedure to obtain energy decay, the main step in proving finite time blow-up results is to choose suitable Lyapunov functional which is closely related to the so-called modified energy functional frequently used in obtaining energy decay, and to employ an convexity argument to obtain with and . This idea would be useful in proving our finite time blow-up result.

More recently, Song [26] established a global nonexistence of solutions with positive initial data for IBVPInspired by [26], Hao and Wei [27] studied IBVPand they proved a global existence and a blow-up criterion for solutions to IBVP (6). Both Refs. [26, 27] assume the same restrictions on , , and . The results obtained by [26, 27] mean that the damping terms and are beaten, in the sense that some solutions blow up in finite time, independently by the source term in the “arena” . We are tempted to compare the damping term and the source term . The later one was shown in a certain condition to be “stronger” than the former one in the arena of viscoelastic plate equation . More, precisely, Li et al. [28] provided a finite time blow-up result for IBVPInspired by [26–28], we seek to prove that some solutions with positive initial data to IBVP (1) blow up in finite time under certain conditions (see Theorem 11); all solutions to IBVP (1) exist globally under other conditions (see Theorem 12). Different from [26], we provide a global existence result under some conditions, in addition to a finite time blow-up result. Compared with the finite time blow-up results in [26, 27], our finite time blow-up result shows that the source beats the combination of the frictional damping and the Kelvin-Voigt damping.

Notations 1. Throughout the paper, is a generic positive constant and it can take a different value at every new occurrence. (resp. ) is the classical Lebsegue (resp., Sobolev) space.

The rest of our paper is planned as follows. In Section 2, we collect the main results and some preliminaries necessary in proving the results. In Section 3, we prove in a certain detail all the main results. In Section 4, we present an open question which confuses us recently.

2. Preliminaries and the Main Results

In this section, we collect the main assumptions on the relaxation function and the parameters in IBVP (1), some results which can simplify our later development and the main results of this paper. Due to their pervasion in the textbooks on PDEs or in the references cited below, all the proofs of the following lemmas are omitted here. Let us recall first Young’s inequality with a .

Lemma 2. For every pair , and every , we have

We list here all the assumptions which are necessary for our claimed results.

Assumption 3. The relaxation function is absolutely continuous and decreasing and satisfies

Assumption 4. ; and if ; and if .

Assumption 5. and .

Assumption 6. and .

Assumption 7. and where is given by (14):

Remark 8. By the well-known Poincaré’s embedding, is a number in .

With the above assumptions prepared, we are now in a position to state the main results of this paper. What comes first is the local existence theory.

Theorem 9. Let Assumptions 3 and 4 be fulfilled. For every pair , there exists a (unique) such that IBVP (1) admits a unique solution, and that is the maximal time interval of existence of .

Remark 10. is called the lifespan of . In the situation , the solution is said to exist globally; in the situation , the solution is said to blow up in finite time.

Our second main result is the following finite time blow-up result.

Theorem 11. Let Assumptions 3, 4, and 5 be fulfilled. Then every solution to IBVP (1) blows up in finite time whenever it satisfies Assumption 7: If satisfies Assumption 7, then we have . And in this situation, we have furthermore

The third main result is the following global existence result.

Theorem 12. Let Assumptions 3, 4, and 6 be fulfilled. Then for every pair , IBVP (1) admits a unique global solution .

In the proofs of the above theorems, the following modified energy functionals are very important:here and hereafter, we write for every

The following lemma would be used several times later.

Lemma 13. Let be defined by (14). Then along the solution to IBVP (1), we have

Remark 14. It should be noted that is decreasing whenever fulfills Assumption 3.

3. Proofs of the Main Results

In this section, we present the proofs of our main results.

Proof of Theorem 9. Since one can prove the local existence by mimicking the steps in [6] and by using the classical Faedo-Galerkin method, the details are omitted here. By combining the local existence result and a standard continuation method, we can show that there exists a maximal time of existence of the solution for every pair .

Before proving Theorem 11, we introduce the following lemma.

Lemma 15 (see [26]). Let Assumptions 3, 4, and 5 be fulfilled. There exists a such that whenever is a solution to IBVP (1) verifying and where is given by

Proof of Theorem 11. Let . We write for the solution to IBVP (1) with its initial datum and write for the maximal time of existence of . Our goal is to show by a contradiction argument that
Assume . Let us introduce the following functional for :where is given by (14) and , and are yet to be determined later. Differentiate both hand sides of (19), and conduct some routine but tedious calculations, to obtainWe next treat and term-by-term. By utilizing the algebraic identity for every , we conclude thatwhere is yet to be selected later. By the definition (14) of , we haveThis, together with (22), impliesSet Thanks to we have, for every , where “” follows from Lemma 15. We fix and such that and thereforeThe existence of and can be justified by the following observation By Lemma 2, for every , By Remark 14 and by the assumption that , we have This, together with (31), impliesCombine (20), (24), (29), and (33), to obtain for , and every and every ,Let , and choose sufficiently small and such thatwhere depends on , , , , , , and . Since the function is convex, by Jensen’s inequality, we have Since , is bounded by some constant . Since is increasing, we obtain On the other hand, we have the following sequence of inequalities:where depends on the measure of ; “” in the first line follows from Hölder’s inequality; “” in the third line follows from the continuous embedding ; “” follows from Hölder’s inequality; “” in the fourth line follows from Young’s inequality (see Lemma 2) and from We fix such that and hence By Remark 14, this implieswhere “” in the third line follows from Combine (38) and (42), to yield This, together with (37), implies directly the following differential inequality: which implies in turnwhere depends on , and . The proof is complete.

Proof of Theorem 12. Let , and let , where is the maximal time of existence, be the solution to IBVP (1) associated with the initial datum . Our objective is to prove . We achieve this goal by contradiction. Assume to the contrary , and we associate the following functional to where is given as in (14). By using (16), we can deduce from Assumption 3 the following inequalityBy Young’s inequality (see Lemma 2), we havewhere is a positive number which would be specified explicitly later and is the best Poincaré’s embedding constant (see (11) for the definition). Plug (49) into (48) and write the result compactly, to yield By choosing , we obtain the following differential inequality which implies immediately where is a constant depending on , and . By the definition (47), we have Now choose such thatwhere is the maximal time of existence of solutions whose initial data’s norms are bounded by . By Theorem 9, IBVP (1) supplemented by the initial condition admits a unique solution with the maximal time of existence of the solution. Write By the construction, is the unique solution to IBVP (1) on associated with the initial datum which is given at the very beginning of the proof. By the definition of , . This contradicts (54).
Therefore, . The proof is complete.

4. Conclusion

In this paper, we studied a class of viscoelastic wave equations which incorporate a pure power source term, a nonlinear frictional damping term, and a strong linear damping term of Kelvin-Voigt type. We proved first a finite time blow-up result by a convexity argument and obtained second a global existence result by a priori estimates for the problem in question. The modified energy functional (14) played an important role in finishing the proofs of the finite time blow-up result and the global existence result.

There remains a problem concerning IBVP (1), that is, the problem of finding or estimating the lifespan for every solution to IBVP (1). When Assumptions 3, 4, and 6 are fulfilled, all the solutions exist globally, and hence their lifespans are . When Assumptions 3, 4, and 5 are fulfilled, the problem of finding lifespans becomes very complicated; in this case, for those initial data satisfying the hypothesis of Theorem 12, we can deduce from (46) the upper bound estimate for the lifespans:where depends on , , and the initial data . In view of this, we are tempted to understand better the following questions: (i)How to obtain an estimate better than (58) upon lifespans?(ii)Let Assumptions 3, 4, and 5 be fulfilled. How to obtain an estimate upon lifespans for solutions whose initial data do not satisfy the hypothesis of Theorem 12?

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Fosheng is supported by the Initial Foundation of Mianyang Teachers’ College (Grant no. QD2016A003). Chengqiang is supported by NSFC (nos. 11701050 and 11571244), by JG Program (no. 2017JG13) of Chengdu Normal University, and by SCJYT Program (no. 18ZB0098) of Sichuan Province, China.