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# A blow up result for viscoelastic equations with arbitrary positive initial energy

- Jie Ma
^{1}Email author, - Chunlai Mu
^{1}and - Rong Zeng
^{1}

**2011**:6

https://doi.org/10.1186/1687-2770-2011-6

© Ma et al; licensee Springer. 2011

**Received: **5 March 2011

**Accepted: **12 July 2011

**Published: **12 July 2011

## Abstract

## Keywords

## 1 Introduction

where Ω is a bounded domain of *R*^{
n
} with smooth boundary ∂Ω, *p* > 1, *q* > 1 and *g* is a positive function. The wave equations (1.1) appear in applications in various areas of mathematical physics (see [1]).

If the equations in (1.1) have not the viscoelastic term
, the equations are known as the wave equation. In this case, the equations have been extensively studied by many people. We observe that the wave equation subject to nonlinear boundary damping has been investigated by the authors Cavalcanti et al. [2, 3] and Vitillaro [4, 5]. It is important to mention other papers in connection with viscoelastic effects such as Aassila et al. [6, 7] and Cavalcanti et al. [8]. Furthermore, related to blow up of the solutions of equations with nonlinear damping and source terms acting in the domain we can cite the work of Alves and Cavalcanti [9], Cavalcanti and Domingos Cavalcanti [10]. As regards non-existence of a global solution, Levine [11] firstly showed that the solutions with negative initial energy are non-global for some abstract wave equation with linear damping. Later Levine and Serrin [12] studied blow-up of a class of more generalized abstract wave equations. Then Pucci and Serrin [13] claimed that the solution blows up in finite time with positive initial energy which is appropriately bounded. In [14] Levine and Todorova proved that there exist some initial data with arbitrary positive initial energy such that the corresponding solution to the wave equations blows up in finite time. Then Todorova and Vitillaro [15] improved the blow-up result above. However, they did not give a sufficient condition for the initial data such that the corresponding solution blows up in finite time with arbitrary positive initial energy. Recently, for problem (1.1) with *g* ≡ 0 and *m* = 1, Gazzalo and Squassina [16] established the condition for initial data with arbitrary positive initial energy such that the corresponding solution blows up in finite time. Zeng et al. [17] studied blowup of solutions for the Kirchhoff type equation with arbitrary positive initial energy.

*g*(

*s*) and

*a*(

*x*). At this point it is important to mention some papers in connection with viscoelastic effects, among them, Alves and Cavalcanti [9], Aassila et al. [7], Cavalcanti and Oquendo [19] and references therein. Then Messaoudi [20] obtained the global existence of solutions for the viscoelastic equation, at same time he also obtained a blow-up result with negative energy. Furthermore, he improved his blow-up result in [21]. Recently, Wang and Wang [22] investigated the following problem

and showed the global existence of the solutions if the initial data are small enough. Moreover, they derived decay estimate for the energy functional. In [23] Wang established the blow-up result for the above problem when the initial energy is high.

We next state some assumptions on *g*(*s*) and real numbers *p* > 1, *q* > 1.

for all *v* ∈ *C*^{1}([0, ∞)) and *t* > 0.

**Remark 1.1**. It is clear that *g*(*t*) = *εe* ^{
-t
} (0 < ε < 1) satisfies the assumptions (A1) and (A2).

Based on the method of Faedo-Galerkin and Banach contraction mapping principle, the local existence and uniqueness of the problem (1.2) have been established in [8, 18, 25, 26] as follows.

**Theorem 1.1**. Under the assumptions (A1)-(A3), let the initial data , (

*u*

_{1},

*v*

_{1}) ∈

*L*

^{2}(Ω) ×

*L*

^{2}(Ω). Then the problem (1.2) has a unique local solution

for the maximum existence time *T*, where *T* ∈ (0, ∞].

Our main blow-up result for the problem (1.2) with arbitrarily positive initial energy is stated as follows.

**Theorem 1.2**. Under the assumptions (A1)-(A3), if and the initial data and (

*u*

_{1},

*v*

_{1}) ∈

*L*

^{2}(Ω) ×

*L*

^{2}(Ω) satisfy

*χ*is the constant of the Poincaré's inequality on Ω, , energy functional

*E*(

*t*) and

*I*(

*u*,

*v*) are defined as

The rest of this paper is organized as follows. In Section 2, we introduce some lemmas needed for the proof of our main results. The proof of our main results is presented in Section 3.

## 2 Preliminaries

In this section, we introduce some lemmas which play a crucial role in proof of our main result in next section.

**Lemma 2.1**. *E*(*t*) is a non-increasing function.

**Lemma 2.2**. Assume that

*g*(

*t*) satisfies assumptions (A1) and (A2),

*H*(

*t*) is a twice continuously differentiable function and satisfies

for every *t* ∈ [0, *T*_{0}), and (*u*(*x*, *t*), *v*(*x*, *t*)) is the solution of the problem (1.2).

Then the function *H*(*t*) is strictly increasing on [0, *T*_{0}).

for every *t* ∈ [0, *T*_{0}).

for every *t* ∈ [0, *T*_{0}).

for every *t* ∈ [0, *T*_{0}).

Because *g*(*t*) satisfies (A2), then *h*'(*t*) ≥ 0, which implies that *h*(*t*) ≥ *h*(0) = *H*(0). Moreover, we see that *H*'(0) > *h*'(0).

*t*

_{0}> 0 and

*H*' (

*t*

_{0}) =

*h*' (

*t*

_{0}), and have

This contradicts *H*'(*t*_{0}) = *h*'(*t*_{0}). Thus, we have *H*'(*t*) > *h*' (*t*) *≥* 0, which implies our desired result. The proof of Lemma 2.2 is complete.

then
is strictly increasing on [0, *T* ).

**Proof**. Since , and (

*u*(

*t*),

*v*(

*t*)) is the local solution of problem (1.2), by a simple computation, we have

Therefore, by Lemma 2.2, the proof of Lemma 2.3 is complete.

**Lemma 2.4**. If , (

*u*

_{1},

*v*

_{1}) ∈

*L*

^{2}(Ω) ×

*L*

^{2}(Ω) satisfy the assumptions in Theorem 1.2, then the solution (

*u*(

*x*,

*t*),

*v*(

*x*,

*t*)) of problem (1.2) satisfies

for every *t* ∈ [0, *T*).

**Proof**. We will prove the lemma by a contradiction argument. First we assume that (2.9) is not true over [0,

*T*), it means that there exists a time

*t*

_{1}such that

*I*(

*u*(

*t*,

*x*),

*v*(

*t*,

*x*)) < 0 on [0,

*t*

_{1}), by Lemma 2.3 we see that is strictly increasing over [0,

*t*

_{1}), which implies

Obviously, (2.15) contradicts to (2.12). Thus, (2.9) holds for every *t* ∈ [0, *T*).

for every *t* ∈ [0, *T*). The proof of Lemma 2.4 is complete.

## 3 The proof of Theorem 1.2

where *t*_{2}, *t*_{3} and *a* are certain positive constants determined later.

which means that *G*"(*t*) > 0 for every *t* ∈ (0, *T*).

Since *G*'(0) ≥ 0 and *G*(0) ≥ 0, thus we obtain that *G*' (*t*) and *G*(*t*) are strictly increasing on [0, *T*).

which implies that *B*^{2} - *AC* ≤ 0.

*u*(

*t*,

*x*),

*v*(

*t*,

*x*)) to the problem (1.2) exists for every

*t*∈ [0,

*T*), then for

*t*∈ [0,

*T*), one has

for every *t* ∈ [0, *T*), which means that the function *G* ^{
-β
} is concave.

The proof of Theorem 1.2 is complete.

## Declarations

### Acknowledgements

This work is supported in part by NSF of PR China (11071266) and in part by Natural Science Foundation Project of CQ CSTC (2010BB9218).

## Authors’ Affiliations

## References

- Fabrizio M, Morro A: Mathematical problems in linear viscoelasticity.
*SIAM Studies in Applied Mathematics Philadelphia*1992., 12:Google Scholar - Cavalcanti MM, Cavalcanti VND, Lasiecka I: Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping source interaction.
*J Diff Equ*2007, 236(2):407-459. 10.1016/j.jde.2007.02.004View ArticleMathSciNetGoogle Scholar - Cavalcanti MM, Cavalcanti VND, Martinez P: Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term.
*J Diff Equ*2004, 203(1):119-158. 10.1016/j.jde.2004.04.011View ArticleMathSciNetGoogle Scholar - Vitillaro E: A potential well theory for the wave equation with nonlinear source and boundary damping terms.
*Glasg Math J*2002, 44(3):375-395. 10.1017/S0017089502030045View ArticleMathSciNetGoogle Scholar - Vitillaro E: Global existence for the wave equation with nonlinear boundary damping and source terms.
*J Diff Equ*2002, 186(1):259-298. 10.1016/S0022-0396(02)00023-2View ArticleMathSciNetGoogle Scholar - Aassila M, Cavalcanti MM, Cavalcanti VND: Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term.
*Calc Var Partial Diff Equ*2002, 15(2):155-180. 10.1007/s005260100096View ArticleMathSciNetGoogle Scholar - Aassila M, Cavalcanti MM, Soriano JA: Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain.
*SIAM J Control Optim*2000, 38(5):1581-1602. 10.1137/S0363012998344981View ArticleMathSciNetGoogle Scholar - Cavalcanti MM, Cavalcanti VND, Soriano JA: Existence and uniform decay rate for viscoelastic problems with nonlinear boundary damping.
*Diff Integ Equ*2001, 14: 85-116.MathSciNetGoogle Scholar - Alves CO, Cavalcanti MM: On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source.
*Calc Var Partial Diff Equ*2009, 34(3):377-411. 10.1007/s00526-008-0188-zView ArticleMathSciNetGoogle Scholar - Cavalcanti MM, Cavalcanti VND: Existence and asymptotic stability for evolution problems on manifolds with damping and source terms.
*J Math Anal Appl*2004, 291(1):109-127. 10.1016/j.jmaa.2003.10.020View ArticleMathSciNetGoogle Scholar - Levine HA: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.
*SIAM J Math Anal*1974, 5: 138-146. 10.1137/0505015View ArticleMathSciNetGoogle Scholar - Levine HA, Serrin J: Global nonexistence theorems for quasilinear evolution equation with dissipation.
*Arch Ration Mech Anal*1997, 137: 341-361. 10.1007/s002050050032View ArticleMathSciNetGoogle Scholar - Pucci P, Serrin J: Global nonexistence for abstract evolution equation with positive initial energy.
*J Diff Equ*1998, 150: 203-214. 10.1006/jdeq.1998.3477View ArticleMathSciNetGoogle Scholar - Levine HA, Todorova G: Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy.
*Proc Am Math Soc*2001, 129: 793-805. 10.1090/S0002-9939-00-05743-9View ArticleMathSciNetGoogle Scholar - Todorova G, Vitillaro E: Blow-up for nonlinear dissipative wave equations in
*R*^{ n }.*J Math Anal Appl*2005, 303: 242-257. 10.1016/j.jmaa.2004.08.039View ArticleMathSciNetGoogle Scholar - Gazzola F, Squassina M: Global solutions and finite time blow up for damed semilinear wave equations.
*Ann Inst H Poincare Anal NonLineaire*2006, 23(2):185-207. 10.1016/j.anihpc.2005.02.007View ArticleMathSciNetGoogle Scholar - Zeng R, Mu CL, Zhou SM: A blow up result for Kirchhoff type equations with high energy.
*Math Methods Appl Sci*2011, 34(4):479-486.MathSciNetGoogle Scholar - Cavalcanti MM, Cavalcanti VND, Soriano JA: Exponential decay for the solution of the semilinear viscoelastic wave equations with localized damping.
*Electron J Diff Equ*2002, 44: 1-14.MathSciNetGoogle Scholar - Cavalcanti MM, Oquendo HP: Frictional versus viscoelastic damping in a semilinear wave equation.
*SIAM J Control Optim*2003, 42(4):1310-1324. 10.1137/S0363012902408010View ArticleMathSciNetGoogle Scholar - Messaoudi SA: Blow up and global existence in a nonlinear viscoelastic wave equation.
*Math Nachr*2003, 260: 58-66. 10.1002/mana.200310104View ArticleMathSciNetGoogle Scholar - Messaoudi SA: Blow up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation.
*J Math Anal Appl*2006, 320: 902-915. 10.1016/j.jmaa.2005.07.022View ArticleMathSciNetGoogle Scholar - Wang YJ, Wang YF: Exponential energy decay of solutions of viscoelastic wave equations.
*J Math Anal Appl*2008, 347: 18-25. 10.1016/j.jmaa.2008.05.098View ArticleMathSciNetGoogle Scholar - Wang YJ: A global nonexistence theorem for viscoelastic equations with arbitrary positive initial energy.
*Appl Math Lett*2009, 22: 1394-1400. 10.1016/j.aml.2009.01.052View ArticleMathSciNetGoogle Scholar - Levine HA: Instability and nonexistence of global solutions of nonlinear wave equation of the form
*Pu*_{ tt }= Δ*u*+*F*(*u*).*Trans Am Math Soc*1974, 192: 1-21.Google Scholar - Cavalcanti MM, Cavalcanti VND, Ferreira J: Existence and uniform decay for nonlinear viscoelastic equation with strong damping.
*Math Meth Appl Sci*2001, 24: 1043-1053. 10.1002/mma.250View ArticleMathSciNetGoogle Scholar - Georgiev V, Todorova G: Existence of a solution of the wave equation with nonlinear damping and source term.
*J Diff Equ*1994, 109: 295-308. 10.1006/jdeq.1994.1051View ArticleMathSciNetGoogle Scholar

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