General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

Liu, Wenjun; Li, Gang; Hong, Linghui

doi:https://doi.org/10.1155/2014/284809

Journal of Function Spaces

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2014 | Article ID 284809 | https://doi.org/10.1155/2014/284809

General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping

Wenjun Liu,¹Gang Li,¹and Linghui Hong¹

Academic Editor: William Ziemer

Received08 May 2013

Revised28 Sept 2013

Accepted16 Nov 2013

Published04 Feb 2014

Abstract

The general decay and blow-up of solutions for a system of viscoelastic equations of Kirchhoff type with strong damping is considered. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy by exploiting the convexity technique, the other is for certain solutions with arbitrarily positive initial energy based on the method of Li and Tsai. Then, we give a decay result of global solutions by the perturbed energy method under a weaker assumption on the relaxation functions.

1. Introduction

In this work, we investigate the following system of viscoelastic equations of Kirchhoff type: where is a bounded domain with smooth boundaryis a positive locally Lipschitz function, andare given functions to be specified later.

To motivate our work, let us recall some previous results regarding viscoelastic equations of Kirchhoff type. The following problem: is a model to describe the motion of deformable solids as hereditary effect is incorporated. It was first studied by Torrejón and Yong [1] who proved the existence of a weakly asymptotic stable solution for large analytical datum. Later, Muñoz Rivera [2] showed the existence of global solutions for small datum and the total energy decays to zero exponentially under some restrictions. Then, Wu and Tsai [3] treated problem (2) for and proved the global existence, decay, and blow-up with suitable conditions on initial data. They obtained the blow-up properties of local solution with small positive initial energy by the direct method of [4]. To obtain the decay result, they assumed that the nonnegative kernelfor someThis energy decay result was recently improved by Wu in [5] under a weaker condition on (i.e.,for ). For a single wave equation of Kirchhoff type without the viscoelastic term, we refer the reader to Matsuyama and Ikehata [6] and Ono [7–10].

Many results concerning local existence, global existence, decay, and blow-up of solutions for a system of wave equations of Kirchhoff type without viscoelastic terms (i.e., ) have also been extensively studied. For example, Park and Bae [11] considered the system of wave equations with nonlinear dampings forand, and showed the global existence and asymptotic behavior of solutions under some restrictions on the initial energy. Later, Benaissa and Messaoudi [12] discussed blow-up properties for negative initial energy. Recently, Wu and Tsai [13] studied the system (1) for. Under some suitable assumptions on they proved local existence of solutions by applying the Banach fixed point theorem and the blow-up of solutions by using the method of Li and Tsai in [4], where three different cases on the sign of the initial energyare considered.

In the case ofand in the presence of viscoelastic term (i.e.,), Cavalcanti et al. [14] studied the equation that was subject to a locally distributed dissipation with the same initial and boundary conditions as that of (2), and proved an exponential decay rate. This work extended the result of Zuazua [15], in which he considered (3) with and the localized linear damping. By using the piecewise multipliers method, Cavalcanti and Oquendo [16] investigated the equation with the same initial and boundary conditions as that of (2). Under the similar conditions on the relaxation functionas above, and for all , they improved the results of [14] by establishing exponential stability for exponential decay functionand linear function and polynomial stability for polynomial decay functionand nonlinear function, respectively.

Concerning blow-up results, Messaoudi [17] considered the equation He proved that any weak solution with negative initial energy blows up in finite time ifand while exists globally for any initial data in the appropriate space if This result was improved by the same author in [18] for positive initial energy under suitable conditions on , and. Recently, Liu [19] studied the equation with the same initial and boundary condition as that of (2). By virtue of convexity technique and supposing that where, he proved that the solution with nonpositive initial energy as well as positive initial energy blows up in finite time.

We should mention that the following system: was considered by Han and Wang in [20], where is a bounded domain with smooth boundary in . Under suitable assumptions on the functions , the initial data and the parameters in the above problem established local existence, global existence, and blow-up property (the initial energy). This latter blow-up result has been improved by Messaoudi and Said-Houari [21] into certain solutions with positive initial energy. Recently, Liang and Gao in [22] investigated the following problem: with the same initial and boundary conditions as that of (9). Under suitable assumptions on the functionsand certain initial data in the stable set, they proved that the decay rate of the solution energy is exponential. Conversely, for certain initial data in the unstable set, they proved that there are solutions with positive initial energy that blow up in finite time. It is also worth mentioning the work [23] in which we studied system (1). Under suitable assumptions on the functionsand certain initial conditions, we showed that the solutions are global in time and the energy decays exponentially. For other papers related to existence, uniform decay, and blow-up of solutions of nonlinear wave equations, we refer the reader to [14, 24–29] for existence and uniform decay, to [17, 30–34] for blow-up, and to [35–40] for the coupled system. To the best of our knowledge, the general decay and blow-up of solutions for systems of viscoelastic equations of Kirchhoff type with strong damping have not been well studied.

Motivated by the above mentioned research, we consider in the present work the coupled system (1) with nonzeroand nonconstant. We note that in such a coupled system case we should overcome the additional difficulties brought by the treatment of the nonlinear coupled terms. We first establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy, the other is for certain solutions with arbitrarily positive initial energy. Then, we give a decay result of global solutions under a weaker assumption on the relaxation functions.

This paper is organized as follows. In the next section we present some assumptions, notations and known results and state the main results: Theorems 4, 5, 6, and 7. The two blow-up results, Theorems 5 and 6, are proved in Sections 3 and 4, respectively. Section 5 is devoted to the proof of the decay result—Theorem 7.

2. Preliminaries and Main Result

In this section we present some assumptions, notations, and known results and state the main results. First, we make the following assumptions.(A1)is a positive locally Lipschitz function forwith the Lipschitz constantsatisfying (A2)are strictly decreasingfunctions such that (A3)There exist two positive differentiable functions and such that andsatisfies (A4)We make the following extra assumption on: where.

Remark 1. It is clear that (which occurs physically in the study of vibrations of damped flexible space structures in a bounded domain in) satisfies (A4) for as long as Indeed, by straightforward calculations, we obtain

Next, we introduce some notations. Consider the Hilbert spaceendowed with the inner product and the functionsand(see also [21]): whereare constants andsatisfies

One can easily verify that where

Then, we give two lemmas which will be used throughout this work.

Lemma 2 (Sobolev-Poincaré inequality [41]). If , then holds with some constant.

Lemma 3 ([21, Lemma 3.2]). Assume that (20) holds. Then there exists such that for any one has

We now state a local existence theorem for system (1), whose proof follows the arguments in [3, 13].

Theorem 4. Suppose that (20), , and hold, and that and Then problem (1) has a unique local solution for someMoreover, at least one of the following statements is valid:

The energy associated with system (1) is given by As in [5], we can get Then we have

We introduce where is the optimal constant in (24).

Our first result is concerned with the blow-up for certain local solutions with nonpositive initial energy as well as positive initial energy.

Theorem 5. Assume that-, and (20) hold. Letbe the unique local solution to system (1). For any fixed, assuming that we can choosesatisfy Suppose further that where , then .

Our second result shows that certain local solutions with arbitrarily positive initial energy can also blow up.

Theorem 6. Suppose that (A1)-(A2), (A4), (20), and hold. Assume further that and and satisfy then the solution of problem (1) blows up at a finite timein the sense of (119) below. Moreover, the upper bounds forcan be estimated by whereandis given in (116) below.

Finally, we state the general decay result. For convenience, we choose especially, and Then the energy functional , defined by (27), becomes

Theorem 7. Suppose that (20),-, and hold, and that and and satisfy and Then for each there exist two positive constantsandsuch that the energy of (1) satisfies where.

To achieve general decay result we will use a Lyapunov type technique for some perturbation energy following the method introduced in [42]. This result improves the one in Li et al. [23] in which only the exponential decay rates are considered.

3. Blow-Up of Solutions with Initial Data in the Unstable Set

In this section, we prove a finite time blow-up result for initial data in the unstable set. We need the following lemmas.

Lemma 8. Suppose that (20),,, andhold. Letbe the solution of system (1). Assume further thatand Then there exists a constantsuch that

Proof. We first note that, by (27), (24) and the definition of we have where we have usedand It is easy to verify that is increasing in , decreasing in , and that , as , and whereis given in (30). Since there exists such that . Set , then from (41) we can get , which implies that .
Now we establish (40) by contradiction. First we assume that (40) is not true over then there existssuch that By the continuity ofwe can choosesuch that Again, the use of (41) leads to This is impossible sincefor all .
Hence (40) is established.

Lemma 9 (see [43, 44]). Let be a positive, twice differentiable function, which satisfies, for , the inequality for some . If and , then there exists a time such that .

Proof of Theorem 5. Assume by contradiction that the solutionis global. Then we consider defined by where , , and are positive constants to be chosen later. Then for all . Furthermore and, consequently, for almost every Testing the first equation of system (1) with and the second equation of system (1) with , integrating the results over , using integration by parts, and summing up, we have which implies Therefore, we have where are the functions defined by Using the Cauchy-Schwarz inequality, we obtain
Similarly, we have By Hlder’s inequality and Young’s inequality, we obtain The previous inequalities imply thatfor everyUsing (53), we get for almost every , where is the map defined by For the fourth term on the right hand side of (59), we have Similarly, Combining (59), (60) with (61), we get Since, we have for any , inserting (63) into (62) and utilizing (27), we have Using (29) and (A4), we have
If , that is, , we choose in (65) and small enough such that . Then by (32), we have
If , that is, , we choose and in (65). Then, we get By (32), we have (notice that due to ) that is, for . Therefore, from the above two inequalities and (A2), we can get Since by Lemma 8, there exists a constant such that which implies It follows from (67) and (73) that
Therefore, by (58), (66), and (74), we obtain for almost everyBy (49), we then choosesufficiently large such that consequently, Then by (76) and (77), we choose large enough so that which ensures that. As , letting , we can select such that . By using Lemma 9, we get . This implies that which is a contradiction. Thus,

4. Blow-Up of Solutions with Arbitrarily Positive Initial Energy

In this section, we prove the second blow-up result (Theorem 6) for solutions with arbitrarily positive initial energy. In order to attain our aim, we need the following three lemmas.

Lemma 10 (see [4]). Let and be a nonnegative function satisfying If then for , where is a constant and is the smallest root of the equation

Lemma 11 (see [4]). If is a nonincreasing function on and satisfies the differential inequality where , and , then there exists a finite time such that and the upper bound of is estimated by where.

For the next lemma, we define

Lemma 12. Assume that the conditions of Theorem 6 hold and letbe a solution of (1), then

Proof. By (86), we have Testing the first equation of system (1) with and testing the second equation of system (1) with and plugging the results into the expression of we obtain By (27), (29), and (90), we get By using Hlder’s inequality and Young’s inequality, we have Similarly, Then taking (92) and (93) into account, we obtain Thus, by (A4) and (33), we get
We note that By Hlder’s inequality and Young’s inequality, we obtain from (96) By Hlder’s inequality and Young’s inequality again, it follows from (86), (88), and (97) that In view of (95) and (98), we have where Let Then satisfies (80) for . By (81), we see that if that is, if which is satisfied by the second hypothesis to Theorem 6, then we get from Lemma 10 that Thus, the proof of Lemma 12 is completed.

In what follows, we find an estimate for the life span ofand prove Theorem 6.

Proof of Theorem 6. Let where and is a certain constant which will be specified later. Then we have where For simplicity, we denote It follows from (88), (96), Hlder’s inequality, and Young’s inequality that By (95), we get Applying (107)–(110), we obtain From (104) and (98) we deduce that By Cauchy-Schwarz inequality, the last term in the above inequality is nonnegative. Hence, we have Therefore by (106) and (113), we get Note that by Lemma 12, for Multiplying (114) by and integrating from to , we obtain where We observe that if Then by Lemma 11, there exists a finite time such that . Moreover, the upper bounds of are estimated by Therefore This completes the proof.

Remark 13. The choice of in (104) is possible provided that .

5. General Decay of Solutions

In this section, we prove the general decay of solutions of system (1). The method of proof is similar to that of [42, Theorem ]. We first state a lemma which is similar to the one first proved by Vitillaro in [33] to study a class of a scalar wave equation.

Lemma 14 ([23, Lemma 3.2]). Suppose that (20), (A1), andhold. Letbe the solution of system (1). Assume further that and Then for all.

The auxiliary functionalsof problem (1) are defined as

Lemma 15. Suppose that (20), (A1), and hold. Let be the solution of system (1). Assume further that and Then for all .

Proof. Sinceand it follows from Lemma 14 and (30) that which implies that for , where we have used (24). Further, by (123), we have From (130) and (36), we deduce that

We note that the following functional was introduced in [42]: where and are some positive constants and Here, we use the same functional (132) but choose in (133) and (134).

Lemma 16. There exist two positive constantsandsuch that

Proof. From Hlder’s inequality, Young’s inequality, Lemma 2, (36), and (125), we deduce Applying Hlder’s inequality, Young’s inequality, and Lemma 2 again, it follows that Similarly, applying Hlder’s inequality, Young’s inequality, and Lemma 2 again, we have It follows from (137), (138), (36), and (126) that If we take and to be sufficiently small, then (135) follows from (132), (136), and (139).

Lemma 17. Under the conditions of Theorem 7, the functional , defined by (133) (with), satisfies

Proof. By using the equations in (1) and setting in (133), we easily see that By Cauchy-Schwarz inequality and Young’s inequality, we have In the same way, we can get Using (141)–(145), we obtain So, Lemma 17 is established.

Lemma 18. Under the conditions of Theorem 7, the functional , defined by (134) (with), satisfies where

Proof. By using the equations in (1) and setin (134), we observe that In what follows we will estimate in (149). By Young’s inequality, (A2), and (125), we get Similarly, we have For in (149), applying (A2), Hlder’s inequality, and Young’s inequality, we deduce Similarly, In the same way, we have By Young’s inequality, Lemma 2, and (125) we see that Hence, we infer that By Young’s inequality and Lemma 2 again, we obtain Combining (149)–(157), we complete the proof of Lemma 18.

Proof of Theorem 7. Sinceare positive, we have, for anydenote By using (28), (132), (140), and (147), a series of computations, for we have We choose,, andsmall enough, such that and; then we can check that We choose,, andso small that (162) remains valid and, further So, we arrive at which yields (if needed, one can choose sufficiently small) where is some positive constant. It follows from (165), (A3), and (28) that That is whereis equivalent todue to (135) andis a positive constant. A simple integration of (167) leads to This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referees and the editors for their useful remarks and comments on an early version of this work. This work was partly supported by the National Natural Science Foundation of China (Grant no. 11301277), the Qing Lan Project of Jiangsu Province, and the Chinese Ministry of Finance Project (Grant no. GYHY200906006).

References

R. Torrejón and J. M. Yong, “On a quasilinear wave equation with memory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 1, pp. 61–78, 1991.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. E. Muñoz Rivera, “Global solution on a quasilinear wave equation with memory,” Unione Matematica Italiana B, vol. 8, no. 2, pp. 289–303, 1994.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
S.-T. Wu and L.-Y. Tsai, “On global existence and blow-up of solutions for an integro-differential equation with strong damping,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 979–1014, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M.-R. Li and L.-Y. Tsai, “Existence and nonexistence of global solutions of some system of semilinear wave equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 8, pp. 1397–1415, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S.-T. Wu, “Exponential energy decay of solutions for an integro-differential equation with strong damping,” Journal of Mathematical Analysis and Applications, vol. 364, no. 2, pp. 609–617, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
T. Matsuyama and R. Ikehata, “On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms,” Journal of Mathematical Analysis and Applications, vol. 204, no. 3, pp. 729–753, 1996.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Ono, “Blowing up and global existence of solutions for some degenerate nonlinear wave equations with some dissipation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 30, no. 7, pp. 4449–4457, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
K. Ono, “Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings,” Journal of Differential Equations, vol. 137, no. 2, pp. 273–301, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
K. Ono, “On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation,” Mathematical Methods in the Applied Sciences, vol. 20, no. 2, pp. 151–177, 1997.
View at: Publisher Site | Google Scholar | MathSciNet
K. Ono, “On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 321–342, 1997.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Y. Park and J. J. Bae, “On existence of solutions of nondegenerate wave equations with nonlinear damping terms,” Nihonkai Mathematical Journal, vol. 9, no. 1, pp. 27–46, 1998.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
A. Benaissa and S. A. Messaoudi, “Blow-up of solutions for the Kirchhoff equation of $q$ -Laplacian type with nonlinear dissipation,” Colloquium Mathematicum, vol. 94, no. 1, pp. 103–109, 2002.
View at: Publisher Site | Google Scholar | MathSciNet
S.-T. Wu and L.-Y. Tsai, “On a system of nonlinear wave equations of Kirchhoff type with a strong dissipation,” Tamkang Journal of Mathematics, vol. 38, no. 1, pp. 1–20, 2007.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano, “Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping,” Electronic Journal of Differential Equations, vol. 2002, no. 44, 14 pages, 2002.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Zuazua, “Exponential decay for the semilinear wave equation with locally distributed damping,” Communications in Partial Differential Equations, vol. 15, no. 2, pp. 205–235, 1990.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Cavalcanti and H. P. Oquendo, “Frictional versus viscoelastic damping in a semilinear wave equation,” SIAM Journal on Control and Optimization, vol. 42, no. 4, pp. 1310–1324, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. A. Messaoudi, “Blow up and global existence in a nonlinear viscoelastic wave equation,” Mathematische Nachrichten, vol. 260, pp. 58–66, 2003.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. A. Messaoudi, “Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation,” Journal of Mathematical Analysis and Applications, vol. 320, no. 2, pp. 902–915, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Liu, “Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source,” Topological Methods in Nonlinear Analysis, vol. 36, no. 1, pp. 153–178, 2010.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
X. Han and M. Wang, “Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 11, pp. 5427–5450, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. A. Messaoudi and B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 277–287, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Liang and H. Gao, “Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping,” Boundary Value Problems, vol. 2011, article 22, 19 pages, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
G. Li, L. Hong, and W. Liu, “Exponential energy decay of solutions for a system of viscoelastic wave equations of Kirchhoff type with strong damping,” Applicable Analysis, vol. 92, no. 5, pp. 1046–1062, 2013.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
D. Andrade and L. H. Fatori, “The nonlinear transmission problem with memory,” Boletim da Sociedade Paranaense de Matemática, vol. 22, no. 1, pp. 106–118, 2004.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. S. Prates Filho, and J. A. Soriano, “Existence and exponential decay for a Kirchhoff-Carrier model with viscosity,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 40–60, 1998.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. M. Cavalcanti, V. N. Domingos Cavalcanti, and M. L. Santos, “Existence and uniform decay rates of solutions to a degenerate system with memory conditions at the boundary,” Applied Mathematics and Computation, vol. 150, no. 2, pp. 439–465, 2004.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
V. Komornik, Exact Controllability and Stabilization, Research in Applied Mathematics, Masson, Paris, France, 1994.
View at: MathSciNet
W. Liu, “General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms,” Journal of Mathematical Physics, vol. 50, no. 11, Article ID 113506, 17 pages, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. Y. Park and J. J. Bae, “Variational inequality for quasilinear wave equations with nonlinear damping terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 50, no. 8, pp. 1065–1083, 2002.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Liu, “General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 6, pp. 1890–1904, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
W. Liu and J. Yu, “On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 6, pp. 2175–2190, 2011.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
B. Said-Houari, “Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms,” Differential and Integral Equations, vol. 23, no. 1-2, pp. 79–92, 2010.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
E. Vitillaro, “Global nonexistence theorems for a class of evolution equations with dissipation,” Archive for Rational Mechanics and Analysis, vol. 149, no. 2, pp. 155–182, 1999.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S.-T. Wu, “Blow-up of solutions for a system of nonlinear wave equations with nonlinear damping,” Electronic Journal of Differential Equations, vol. 2009, no. 105, 11 pages, 2009.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
K. Agre and M. A. Rammaha, “Systems of nonlinear wave equations with damping and source terms,” Differential and Integral Equations, vol. 19, no. 11, pp. 1235–1270, 2006.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
M.-R. Li and L.-Y. Tsai, “On a system of nonlinear wave equations,” Taiwanese Journal of Mathematics, vol. 7, no. 4, pp. 557–573, 2003.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
W. Liu, “Uniform decay of solutions for a quasilinear system of viscoelastic equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 5-6, pp. 2257–2267, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
F. Sun and M. Wang, “Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 4, pp. 739–761, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S.-T. Wu, “On decay and blow-up of solutions for a system of nonlinear wave equations,” Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 360–377, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. Yu, “Polynomial stability of solutions for a system of non-linear viscoelastic equations,” Applicable Analysis, vol. 88, no. 7, pp. 1039–1051, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1975.
View at: MathSciNet
S. A. Messaoudi, “General decay of the solution energy in a viscoelastic equation with a nonlinear source,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2589–2598, 2008.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $P u_{t t} = - A u + ℱ (u)$ ,” Archive for Rational Mechanics and Analysis, vol. 51, pp. 371–386, 1973.
View at: Google Scholar | Zentralblatt MATH | MathSciNet
H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form $P u_{t t} = - A u + ℱ (u)$ ,” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974.
View at: Google Scholar | Zentralblatt MATH | MathSciNet

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