Abstract and Applied Analysis

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Recent Progress in Differential and Difference Equations

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Volume 2011 |Article ID 760209 | https://doi.org/10.1155/2011/760209

Liang Fei, Gao Hongjun, "Global Nonexistence of Positive Initial-Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms", Abstract and Applied Analysis, vol. 2011, Article ID 760209, 14 pages, 2011. https://doi.org/10.1155/2011/760209

Global Nonexistence of Positive Initial-Energy Solutions for Coupled Nonlinear Wave Equations with Damping and Source Terms

Academic Editor: Josef DiblΓ­k
Received26 Nov 2010
Revised05 Jun 2011
Accepted27 Jun 2011
Published11 Aug 2011


This work is concerned with a system of nonlinear wave equations with nonlinear damping and source terms acting on both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.

1. Introduction

In this paper we study the initial-boundary-value problem 𝑒𝑑𝑑𝑔||||βˆ’divβˆ‡π‘’2+||π‘’βˆ‡π‘’π‘‘||π‘šβˆ’1𝑒𝑑=𝑓1𝑣(𝑒,𝑣),(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇),𝑑𝑑𝑔||||βˆ’divβˆ‡π‘£2+||π‘£βˆ‡π‘£π‘‘||π‘Ÿβˆ’1𝑣𝑑=𝑓2(𝑒,𝑣),(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇),𝑒(π‘₯,𝑑)=𝑣(π‘₯,𝑑)=0,π‘₯βˆˆπœ•Ξ©Γ—(0,𝑇),𝑒(π‘₯,0)=𝑒0(π‘₯),𝑒𝑑(π‘₯,0)=𝑒1(π‘₯),π‘₯∈Ω,𝑣(π‘₯,0)=𝑣0(π‘₯),𝑣𝑑(π‘₯,0)=𝑣1(π‘₯),π‘₯∈Ω,(1.1) where Ξ© is a bounded domain in ℝ𝑛 with a smooth boundary πœ•Ξ©, π‘š,π‘Ÿβ‰₯1, and 𝑓𝑖(β‹…,β‹…)βˆΆβ„2→ℝ (𝑖=1,2) are given functions to be specified later. We assume that 𝑔 is a function which satisfies π‘”βˆˆπΆ1,𝑔(𝑠)>0,𝑔(𝑠)+2π‘ π‘”ξ…ž(𝑠)>0(1.2) for 𝑠>0.

To motivate our work, let us recall some results regarding 𝑔≑1. The single-wave equation of the form π‘’π‘‘π‘‘ξ€·π‘’βˆ’Ξ”π‘’+β„Žπ‘‘ξ€Έ=𝑓(𝑒),π‘₯∈Ω,𝑑>0(1.3) in Ω×(0,∞) with initial and boundary conditions has been extensively studied, and many results concerning global existence, blow-up, energy decay have been obtained. In the absence of the source term, that is, (𝑓=0), it is well known that the damping term β„Ž(𝑒𝑑) assures global existence and decay of the solution energy for arbitrary initial data (see [1]). In the absence of the damping term, the source term causes finite time blow-up of solutions with a large initial data (negative initial energy) (see [2, 3]). The interaction between the damping term and the source term makes the problem more interesting. This situation was first considered by Levine [4, 5] in the linear damping case β„Ž(𝑒𝑑)=π‘Žπ‘’π‘‘ and a polynomial source term of the form 𝑓(𝑒)=𝑏|𝑒|π‘βˆ’2𝑒. He showed that solutions with negative initial energy blow up in finite time. The main tool used in [4, 5] is the β€œconcavity method.” Georgiev and Todorova in [6] extended Levine's result to the nonlinear damping case β„Ž(𝑒𝑑)=π‘Ž|𝑒𝑑|π‘šβˆ’2𝑒𝑑. In their work, the authors considered problem (1.3) with 𝑓(𝑒)=𝑏|𝑒|π‘βˆ’2𝑒 and introduced a method different from the one known as the concavity method and showed that solutions with negative energy continue to exist globally in time if π‘šβ‰₯𝑝β‰₯2 and blow up in finite time if 𝑝>π‘šβ‰₯2 and the initial energy is sufficiently negative. This latter result has been pushed by Messaoudi [7] to the situation where the initial energy 𝐸(0)<0 and has been improved by the same author in [8] to accommodate certain solutions with positive initial energy.

In the case of 𝑔 being a given nonlinear function, the following equation: π‘’π‘‘π‘‘ξ€·π‘’βˆ’π‘”π‘₯ξ€Έπ‘₯βˆ’π‘’π‘₯π‘₯𝑑||𝑒+𝛿𝑑||π‘šβˆ’1𝑒𝑑=πœ‡|𝑒|π‘βˆ’1𝑒,π‘₯∈(0,1),𝑑>0,(1.4) with initial and boundary conditions has been extensively studied. Equation of type of (1.4) is a class of nonlinear evolution governing the motion of a viscoelastic solid composed of the material of the rate type, see [9–12]. It can also be seen as field equation governing the longitudinal motion of a viscoelastic bar obeying the nonlinear Voigt model, see [13]. In two- and three-dimensional cases, they describe antiplane shear motions of viscoelastic solids. We refer to [14–16] for physical origins and derivation of mathematical models of motions of viscoelastic media and only recall here that, in applications, the unknown 𝑒 naturally represents the displacement of the body relative to a fixed reference configuration. When 𝛿=πœ‡=0, there have been many impressive works on the global existence and other properties of solutions of (1.4), see [9, 10, 17, 18]. Especially, in [19] the authors have proved the global existence and uniqueness of the generalized and classical solution for the initial boundary value problem (1.4) when we replace 𝛿|𝑒𝑑|π‘šβˆ’1𝑒𝑑 and πœ‡|𝑒|π‘βˆ’1𝑒 by 𝑔(𝑒𝑑) and 𝑓(𝑒), respectively. But about the blow-up of the solution for problem, in this paper there has not been any discussion. Chen et al. [20] considered problem (1.4) and first established an ordinary differential inequality, next given the sufficient conditions of blow-up of the solution of (1.4) by the inequality. In [21], Hao et al. considered the single-wave equation of the form 𝑒𝑑𝑑𝑔||||βˆ’divβˆ‡π‘’2ξ‚ξ‚ξ€·π‘’βˆ‡π‘’+β„Žπ‘‘ξ€Έ=𝑓(𝑒),π‘₯∈Ω,𝑑>0(1.5) with initial and Dirichlet boundary condition, where 𝑔 satisfies condition (1.2) and 𝑔(𝑠)β‰₯𝑏1+𝑏2π‘ π‘ž,π‘žβ‰₯0.(1.6) The damping term has the form β„Žξ€·π‘’π‘‘ξ€Έ=𝑑1𝑒𝑑+𝑑2||𝑒𝑑||π‘Ÿβˆ’1𝑒𝑑,π‘Ÿ>1.(1.7) The source term is 𝑓(𝑒)=π‘Ž1𝑒+π‘Ž2|𝑒|π‘βˆ’1𝑒(1.8) with 𝑝β‰₯1 for 𝑛=1,2 and 1≀𝑛≀2𝑛/(π‘›βˆ’2) for 𝑛β‰₯3, π‘Ž1,π‘Ž2,𝑏1,𝑏2,𝑑1,𝑑2 are nonnegative constants, and 𝑏1+𝑏2>0. By using the energy compensation method [7, 8, 22], they proved that under some conditions on the initial value and the growth orders of the nonlinear strain term, the damping term, and the source term, the solution to problem (1.5) exists globally and blows up in finite time with negative initial energy, respectively.

Some special cases of system (1.1) arise in quantum field theory which describe the motion of charged mesons in an electromagnetic field, see [23, 24]. Recently, some of the ideas in [6, 22] have been extended to study certain systems of wave equations. Agre and Rammaha [25] studied the system of (1.1) with 𝑔≑1 and proved several results concerning local and global existence of a weak solution and showed that any weak solution with negative initial energy blows up in finite time, using the same techniques as in [6]. This latter blow-up result has been improved by Said-Houari [26] by considering a larger class of initial data for which the initial energy can take positive values. Recently, Wu et al. [27] considered problem (1.1) with the nonlinear functions 𝑓1(𝑒,𝑣) and 𝑓2(𝑒,𝑣) satisfying appropriate conditions. They proved under some restrictions on the parameters and the initial data several results on global existence of a weak solution. They also showed that any weak solution with initial energy 𝐸(0)<0 blows up in finite time.

In this paper, we also consider problem (1.1) and improve the global nonexistence result obtained in [27], for a large class of initial data in which our initial energy can take positive values. The main tool of the proof is a technique introduced by Payne and Sattinger [28] and some estimates used firstly by Vitillaro [29], in order to study a class of a single-wave equation.

2. Preliminaries and Main Result

First, let us introduce some notation used throughout this paper. We denote by ||β‹…||π‘ž the πΏπ‘ž(Ξ©) norm for 1β‰€π‘žβ‰€βˆž and by ||βˆ‡β‹…||2 the Dirichlet norm in 𝐻10(Ξ©) which is equivalent to the 𝐻1(Ξ©) norm. Moreover, we set (ξ€œπœ‘,πœ“)=Ξ©πœ‘(π‘₯)πœ“(π‘₯)𝑑π‘₯(2.1) as the usual 𝐿2(Ξ©) inner product.

Concerning the functions 𝑓1(𝑒,𝑣) and 𝑓2(𝑒,𝑣), we take 𝑓1ξ€Ί(𝑒,𝑣)=π‘Ž|𝑒+𝑣|2(𝑝+1)(𝑒+𝑣)+𝑏|𝑒|𝑝𝑒|𝑣|(𝑝+2)ξ€»,𝑓2ξ€Ίπ‘Ž(𝑒,𝑣)=|𝑒+𝑣|2(𝑝+1)(𝑒+𝑣)+𝑏|𝑒|(𝑝+2)|𝑣|𝑝𝑣,(2.2) where π‘Ž,𝑏>0 are constants and 𝑝 satisfies ⎧βŽͺ⎨βŽͺβŽ©π‘>βˆ’1,if𝑛=1,2,βˆ’1<𝑝≀4βˆ’π‘›π‘›βˆ’2,if𝑛β‰₯3.(2.3) One can easily verify that 𝑒𝑓1(𝑒,𝑣)+𝑣𝑓2(𝑒,𝑣)=2(𝑝+2)𝐹(𝑒,𝑣),βˆ€(𝑒,𝑣)βˆˆβ„2,(2.4) where 1𝐹(𝑒,𝑣)=2ξ€Ί(𝑝+2)π‘Ž|𝑒+𝑣|2(𝑝+2)+2𝑏|𝑒𝑣|𝑝+2ξ€».(2.5)

We have the following result.

Lemma 2.1 (see [30, Lemma  2.1]). There exist two positive constants 𝑐0 and 𝑐1 such that 𝑐0ξ€·2(𝑝+2)|𝑒|2(𝑝+2)+|𝑣|2(𝑝+2)𝑐≀𝐹(𝑒,𝑣)≀1ξ€·2(𝑝+2)|𝑒|2(𝑝+2)+|𝑣|2(𝑝+2)ξ€Έ.(2.6)

Throughout this paper, we define 𝑔 by 𝑔(𝑠)=𝑏1+𝑏2π‘ π‘ž,π‘žβ‰₯0,𝑏1+𝑏2>0,(2.7) where 𝑏1,𝑏2 are nonnegative constants. Obviously, 𝑔 satisfies conditions (1.2) and (1.6). Set ξ€œπΊ(𝑠)=𝑠0𝑔(𝑠)𝑑𝑠,𝑠β‰₯0.(2.8)

In order to state and prove our result, we introduce the following function space: 𝑍=(𝑒,𝑣)βˆ£π‘’,π‘£βˆˆπΏβˆžξ‚€[0,𝑇);π‘Š01,2(π‘ž+1)(Ξ©)∩𝐿2(𝑝+2),𝑒(Ξ©)π‘‘βˆˆπΏβˆžξ€·[0,𝑇);𝐿2ξ€Έ(Ξ©)βˆ©πΏπ‘š+1𝑣(Ω×(0,𝑇)),π‘‘βˆˆπΏβˆžξ€·[0,𝑇);𝐿2ξ€Έ(Ξ©)βˆ©πΏπ‘Ÿ+1(Ω×(0,𝑇)),𝑒𝑑𝑑,π‘£π‘‘π‘‘βˆˆπΏβˆžξ€·[0,𝑇),𝐿2.(Ξ©)(2.9)

Define the energy functional 𝐸(𝑑) associated with our system 1𝐸(𝑑)=2‖‖𝑒𝑑(‖‖𝑑)22+‖‖𝑣𝑑(‖‖𝑑)22+12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2ξ€œξ‚ξ‚π‘‘π‘₯βˆ’Ξ©πΉ(𝑒,𝑣)𝑑π‘₯.(2.10) A simple computation gives 𝑑𝐸(𝑑)𝑑𝑑=βˆ’β€–π‘’β€–π‘š+1π‘š+1βˆ’β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1≀0.(2.11)

Our main result reads as follows.

Theorem 2.2. Assume that (2.3) holds. Assume further that 2(𝑝+2)>max{2π‘ž+2,π‘š+1,π‘Ÿ+1}. Then any solution of (1.1) with initial data satisfying ξ‚΅ξ€œΞ©ξ‚€πΊξ‚€||βˆ‡π‘’0||2||+πΊβˆ‡π‘£0||2𝑑π‘₯1/2>𝛼1,𝐸(0)<𝐸2,(2.12) cannot exist for all time, where the constant 𝛼1 and 𝐸2 are defined in (3.7).

3. Proof of Theorem 2.2

In this section, we deal with the blow-up of solutions of the system (1.1). Before we prove our main result, we need the following lemmas.

Lemma 3.1. Let Θ(𝑑) be a solution of the ordinary differential inequality π‘‘Ξ˜(𝑑)𝑑𝑑β‰₯𝐢Θ1+πœ€(𝑑),𝑑>0,(3.1) where πœ€>0. If Θ(0)>0, then the solution ceases to exist for 𝑑β‰₯Ξ˜βˆ’πœ€(0)πΆβˆ’1πœ€βˆ’1.

Lemma 3.2. Assume that (2.3) holds. Then there exists πœ‚>0 such that for any (𝑒,𝑣)βˆˆπ‘, one has ‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2ξ‚΅ξ€œβ‰€πœ‚Ξ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝑑π‘₯𝑝+2.(3.2)

Proof. By using Minkowski's inequality, we get ‖𝑒+𝑣‖22(𝑝+2)‖≀2𝑒‖22(𝑝+2)+‖𝑣‖22(𝑝+2).(3.3) Also, HΓΆlder's and Young's inequalities give us ‖𝑒𝑣‖𝑝+2≀‖𝑒‖2(𝑝+2)‖𝑣‖2(𝑝+2)≀12‖𝑒‖22(𝑝+2)+‖𝑣‖22(𝑝+2).(3.4) If 𝑏1>0, then we have ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝑑π‘₯β‰₯π‘β€–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έ.(3.5) If 𝑏1=0, from 𝑏1+𝑏2>0, we have 𝑏2>0. Since π‘Š01,2(π‘ž+1)(Ξ©)β†ͺ𝐻10(Ξ©), we have β€–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22≀𝑐1ξ‚€β€–βˆ‡π‘’β€–22(π‘ž+1)+β€–βˆ‡π‘£β€–22(π‘ž+1),(3.6) which implies that (3.5) still holds for 𝑏1=0. Combining (3.3), (3.4) with (3.5) and the embedding 𝐻10(Ξ©)β†ͺ𝐿2(𝑝+2)(Ξ©), we have (3.2).

In order to prove our result and for the sake of simplicity, we take π‘Ž=𝑏=1 and introduce the following: 𝐡=πœ‚1/(2(𝑝+2)),𝛼1=π΅βˆ’(𝑝+2/(𝑝+1)),𝐸1=ξ‚΅12βˆ’1𝛼2(𝑝+2)21,𝐸2=ξ‚΅1βˆ’12(π‘ž+1)𝛼2(𝑝+2)21,(3.7) where πœ‚ is the optimal constant in (3.2). The following lemma will play an essential role in the proof of our main result, and it is similar to a lemma used first by Vitillaro [29].

Lemma 3.3. Assume that (2.3) holds. Let (𝑒,𝑣)βˆˆπ‘ be the solution of the system (1.1). Assume further that 𝐸(0)<𝐸1 and ξ‚΅ξ€œΞ©ξ‚€πΊξ‚€||βˆ‡π‘’0||2||+πΊβˆ‡π‘£0||2𝑑π‘₯1/2>𝛼1.(3.8) Then there exists a constant 𝛼2>𝛼1 such that ξ‚΅ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝑑π‘₯1/2β‰₯𝛼2ξ‚€,for𝑑>0,(3.9)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+21/(2(𝑝+2))β‰₯𝐡𝛼2,for𝑑>0.(3.10)

Proof. We first note that, by (2.10), (3.2), and the definition of 𝐡, we have 1𝐸(𝑑)β‰₯2ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£21𝑑π‘₯βˆ’ξ‚€2(𝑝+2)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2β‰₯12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝐡𝑑π‘₯βˆ’2(𝑝+2)ξ‚΅ξ€œ2(𝑝+2)Ω𝐺||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝑑π‘₯𝑝+2=12𝛼2βˆ’π΅2(𝑝+2)𝛼2(𝑝+2)2(𝑝+2),(3.11) where βˆ«π›Ό=(Ξ©(𝐺(|βˆ‡π‘’|2)+𝐺(|βˆ‡π‘£|2))𝑑π‘₯)1/2. It is not hard to verify that 𝑔 is increasing for 0<𝛼<𝛼1, decreasing for 𝛼>𝛼1, 𝑔(𝛼)β†’βˆ’βˆž as 𝛼→+∞, and 𝑔𝛼1ξ€Έ=12𝛼21βˆ’π΅2(𝑝+2)𝛼2(𝑝+2)12(𝑝+2)=𝐸1,(3.12) where 𝛼1 is given in (3.7). Since 𝐸(0)<𝐸1, there exists 𝛼2>𝛼1 such that 𝑔(𝛼2)=𝐸(0).
Set 𝛼0∫=(Ξ©(𝐺(|βˆ‡π‘’0|2)+𝐺(|βˆ‡π‘£0|2))𝑑π‘₯)1/2. Then by (3.11) we get 𝑔(𝛼0)≀𝐸(0)=𝑔(𝛼2), which implies that 𝛼0β‰₯𝛼2. Now, to establish (3.9), we suppose by contradiction that ξ‚΅ξ€œΞ©ξ‚€πΊξ‚€||ξ€·π‘‘βˆ‡π‘’0ξ€Έ||2||𝑑+πΊβˆ‡π‘£0ξ€Έ||2𝑑π‘₯1/2<𝛼2,(3.13) for some 𝑑0>0. By the continuity of ∫Ω(𝐺(|βˆ‡π‘’|2)+𝐺(|βˆ‡π‘£|2))𝑑π‘₯, we can choose 𝑑0 such that ξ‚΅ξ€œΞ©ξ‚€πΊξ‚€||ξ€·π‘‘βˆ‡π‘’0ξ€Έ||2||𝑑+πΊβˆ‡π‘£0ξ€Έ||2𝑑π‘₯1/2>𝛼1.(3.14) Again, the use of (3.11) leads to 𝐸𝑑0ξ€Έξƒ©ξ‚΅ξ€œβ‰₯𝑔Ω𝐺||ξ€·π‘‘βˆ‡π‘’0ξ€Έ||2||𝑑+πΊβˆ‡π‘£0ξ€Έ||2𝑑π‘₯1/2ξƒͺ𝛼>𝑔2ξ€Έ=𝐸(0).(3.15) This is impossible since 𝐸(𝑑)≀𝐸(0) for all π‘‘βˆˆ[0,𝑇). Hence (3.9) is established.
To prove (3.10), we make use of (2.10) to get 12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£21𝑑π‘₯≀𝐸(0)+ξ‚€2(𝑝+2)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2.(3.16) Consequently, (3.9) yields 1ξ‚€2(𝑝+2)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2β‰₯12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2β‰₯1𝑑π‘₯βˆ’πΈ(0)2𝛼221βˆ’πΈ(0)β‰₯2𝛼22ξ€·π›Όβˆ’π‘”2ξ€Έ=𝐡2(𝑝+2)2𝛼(𝑝+2)22(𝑝+2).(3.17) Therefore, (3.17) and (3.7) yield the desired result.

Proof of Theorem 2.2. We suppose that the solution exists for all time and we reach to a contradiction. Set 𝐻(𝑑)=𝐸2βˆ’πΈ(𝑑).(3.18) By using (2.10) and (3.18), we have 0<𝐻(0)≀𝐻(𝑑)=𝐸2βˆ’12‖‖𝑒𝑑(‖‖𝑑)22+‖‖𝑣𝑑(‖‖𝑑)22ξ‚βˆ’12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2+1𝑑π‘₯ξ‚€2(𝑝+2)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2.(3.19) From (3.9), we have 𝐸2βˆ’12‖‖𝑒𝑑(‖‖𝑑)22+‖‖𝑣𝑑(‖‖𝑑)22ξ‚βˆ’12ξ€œΞ©ξ‚€πΊξ‚€||||βˆ‡π‘’2||||+πΊβˆ‡π‘£2𝑑π‘₯≀𝐸2βˆ’12𝛼21≀𝐸1βˆ’12𝛼211=βˆ’π›Ό2(𝑝+2)21<0,βˆ€π‘‘β‰₯0.(3.20) Hence, by the above inequality and (2.6), we have 10<𝐻(0)≀𝐻(𝑑)≀2ξ‚€(𝑝+2)‖𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2,≀𝑐(3.21)1ξ‚€2(𝑝+2)‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.22) We then define Θ(𝑑)=𝐻1βˆ’π›Ώ(ξ€œπ‘‘)+πœ–Ξ©ξ€·π‘’π‘’π‘‘+𝑣𝑣𝑑𝑑π‘₯,(3.23) where πœ– small enough is to be chosen later and ξ‚»0<𝛿≀min𝑝+1,2(𝑝+2)2(𝑝+2)βˆ’(π‘š+1),2π‘š(𝑝+2)2(𝑝+2)βˆ’(π‘Ÿ+1)ξ‚Ό2π‘Ÿ(𝑝+2).(3.24) Our goal is to show that Θ(𝑑) satisfies the differential inequality (3.1) which leads to a blow-up in finite time. By taking a derivative of (3.23), we get Ξ˜ξ…ž(𝑑)=(1βˆ’π›Ώ)π»βˆ’π›Ώ(𝑑)π»ξ…ž(‖‖𝑒𝑑)+πœ–π‘‘β€–β€–22+‖‖𝑣𝑑‖‖22ξ‚ξ€œβˆ’πœ–Ξ©ξ‚€π‘”ξ‚€||||βˆ‡π‘’2||||βˆ‡π‘’2ξ‚€||||+π‘”βˆ‡π‘£2||||βˆ‡π‘£2ξ‚ξ€œπ‘‘π‘₯βˆ’πœ–Ξ©ξ‚€||𝑒𝑑||π‘šβˆ’1𝑒𝑑||𝑣𝑒+𝑑||π‘Ÿβˆ’1π‘£π‘‘π‘£ξ‚ξ€œπ‘‘π‘₯+πœ–Ξ©ξ€·π‘’π‘“1(𝑒,𝑣)+𝑣𝑓2ξ€Έ(𝑒,𝑣)𝑑π‘₯=(1βˆ’π›Ώ)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–π‘‘β€–β€–22+‖‖𝑣𝑑‖‖22ξ‚βˆ’π‘1πœ–ξ€·β€–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έβˆ’πœ–π‘2β€–βˆ‡π‘’β€–2(π‘ž+2)2(π‘ž+2)βˆ’πœ–π‘2β€–βˆ‡π‘£β€–2(π‘ž+2)2(π‘ž+2)ξ€œβˆ’πœ–Ξ©ξ‚€||𝑒𝑑||π‘šβˆ’1𝑒𝑑||𝑣𝑒+𝑑||π‘Ÿβˆ’1𝑣𝑑𝑣𝑑π‘₯+πœ–β€–π‘’+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2.(3.25) From the definition of 𝐻(𝑑), it follows that βˆ’π‘2β€–βˆ‡π‘’β€–2(π‘ž+2)2(π‘ž+2)βˆ’π‘2β€–βˆ‡π‘£β€–2(π‘ž+2)2(π‘ž+2)=2(π‘ž+1)𝐻(𝑑)βˆ’2(π‘ž+1)𝐸2‖‖𝑒+(π‘ž+1)𝑑‖‖22+‖‖𝑣𝑑‖‖22+(π‘ž+1)𝑏1ξ€·β€–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έξ€œβˆ’2(π‘ž+1)Ω𝐹(𝑒,𝑣)𝑑π‘₯,(3.26) which together with (3.25) gives Ξ˜β€²(𝑑)=(1βˆ’π›Ώ)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+𝑏1ξ€·β€–π‘žπœ–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έξ€œβˆ’πœ–Ξ©ξ‚€||𝑒𝑑||π‘šβˆ’1𝑒𝑑||𝑣𝑒+𝑑||π‘Ÿβˆ’1𝑣𝑑𝑣𝑑π‘₯+πœ–1βˆ’π‘ž+1‖𝑝+2𝑒+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2+2(π‘ž+1)𝐻(𝑑)βˆ’2(π‘ž+1)𝐸2.(3.27) Then, using (3.10), we obtain Ξ˜β€²(𝑑)β‰₯(1βˆ’π›Ώ)π»βˆ’π›Ώξ‚€β€–β€–π‘’(𝑑)𝐻′(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+𝑏1ξ€·β€–π‘žπœ–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έ+2(π‘ž+1)𝐻(𝑑)+πœ–π‘ξ‚€β€–π‘’+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2ξ‚ξ€œβˆ’πœ–Ξ©ξ‚€||𝑒𝑑||π‘šβˆ’1𝑒𝑑||𝑣𝑒+𝑑||π‘Ÿβˆ’1𝑣𝑑𝑣𝑑π‘₯,(3.28) where 𝑐=1βˆ’(π‘ž+1)/(𝑝+2)βˆ’2(π‘ž+1)𝐸2(𝐡𝛼2)βˆ’2(𝑝+2). It is clear that 𝑐>0, since 𝛼2>π΅βˆ’(𝑝+2)/(𝑝+1). We now exploit Young's inequality to estimate the last two terms on the right side of (3.28) ||||ξ€œΞ©||𝑒𝑑||π‘šβˆ’1𝑒𝑑||||β‰€πœ‚π‘’π‘‘π‘₯1π‘š+1π‘š+1β€–π‘’β€–π‘š+1π‘š+1+π‘šπœ‚1βˆ’((π‘š+1)/π‘š)β€–β€–π‘’π‘š+1π‘‘β€–β€–π‘š+1π‘š+1,||||ξ€œΞ©||𝑣𝑑||π‘Ÿβˆ’1𝑣𝑑||||β‰€πœ‚π‘£π‘‘π‘₯2π‘Ÿ+1π‘Ÿ+1β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1+π‘Ÿπœ‚2βˆ’((π‘Ÿ+1)/π‘Ÿ)β€–β€–π‘£π‘Ÿ+1π‘‘β€–β€–π‘Ÿ+1π‘Ÿ+1,(3.29) where πœ‚1,πœ‚2 are parameters depending on the time 𝑑 and specified later. Inserting the last two estimates into (3.28), we have Ξ˜ξ…ž(𝑑)β‰₯(1βˆ’π›Ώ)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+𝑏1ξ€·β€–π‘žπœ–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έ+2(π‘ž+1)𝐻(𝑑)+πœ–π‘ξ‚€β€–π‘’+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑒𝑣‖𝑝+2𝑝+2ξ‚πœ‚βˆ’πœ–1π‘š+1π‘š+1β€–π‘’β€–π‘š+1π‘š+1βˆ’πœ–π‘šπœ‚1βˆ’((π‘š+1)/π‘š)β€–β€–π‘’π‘š+1π‘‘β€–β€–π‘š+1π‘š+1πœ‚βˆ’πœ–2π‘Ÿ+1β€–π‘Ÿ+1π‘£β€–π‘Ÿ+1π‘Ÿ+1βˆ’πœ–π‘Ÿπœ‚2βˆ’((π‘Ÿ+1)/π‘Ÿ)β€–β€–π‘£π‘Ÿ+1π‘‘β€–β€–π‘Ÿ+1π‘Ÿ+1.(3.30) By choosing πœ‚1 and πœ‚2 such that πœ‚1βˆ’(π‘š+1)/π‘š=𝑀1π»βˆ’π›Ώ(𝑑),πœ‚2βˆ’(π‘Ÿ+1)/π‘Ÿ=𝑀2π»βˆ’π›Ώ(𝑑),(3.31) where 𝑀1 and 𝑀2 are constants to be fixed later. Thus, by using (2.6) and (3.31), inequality (3.31) then takes the form Ξ˜ξ…ž(𝑑)β‰₯((1βˆ’π›Ώ)βˆ’π‘€πœ–)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+𝑏1ξ€·β€–π‘žπœ–βˆ‡π‘’β€–22+β€–βˆ‡π‘£β€–22ξ€Έ+2(π‘ž+1)𝐻(𝑑)+πœ–π‘2‖𝑒‖2(𝑝+2)2(𝑝+2)+2‖𝑣‖2(𝑝+2)2(𝑝+2)ξ‚βˆ’πœ–π‘€1βˆ’π‘šπ»π›Ώπ‘š(𝑑)β€–π‘’β€–π‘š+1π‘š+1βˆ’πœ–π‘€2βˆ’π‘Ÿπ»π›Ώπ‘Ÿ(𝑑)β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1,(3.32) where 𝑀=π‘š/(π‘š+1)𝑀1+π‘Ÿ/(π‘Ÿ+1)𝑀2 and 𝑐2 is a positive constant.
Since 2(𝑝+2)>max{π‘š+1,π‘Ÿ+1}, taking into account (2.6) and (3.21), then we have π»π›Ώπ‘šβ€–(𝑑)π‘’β€–π‘š+1π‘š+1≀𝑐3‖𝑒‖2π›Ώπ‘š(𝑝+2)+(π‘š+1)2(𝑝+2)+‖𝑣‖2π›Ώπ‘š(𝑝+2)2(𝑝+2)β€–π‘’β€–π‘š+1π‘š+1,π»π›Ώπ‘Ÿ(𝑑)β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1≀𝑐4‖𝑣‖2π›Ώπ‘Ÿ(𝑝+2)+(π‘Ÿ+1)2(𝑝+2)+‖𝑒‖2π›Ώπ‘Ÿ(𝑝+2)2(𝑝+2)β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1,(3.33) for some positive constants 𝑐3 and 𝑐4. By using (3.24) and the algebraic inequality π‘§πœˆξ‚€1≀𝑧+1≀1+π‘Žξ‚(𝑧+π‘Ž),βˆ€π‘§β‰₯0,0<πœˆβ‰€1,π‘Žβ‰₯0,(3.34) we have ‖𝑒‖2π›Ώπ‘š(𝑝+2)+(π‘š+1)2(𝑝+2)‖≀𝑑𝑒‖2(𝑝+2)2(𝑝+2)‖+𝐻(0)≀𝑑𝑒‖2(𝑝+2)2(𝑝+2)+𝐻(𝑑),βˆ€π‘‘β‰₯0,(3.35) where 𝑑=1+1/𝐻(0). Similarly, ‖𝑣‖2π›Ώπ‘Ÿ(𝑝+2)+(π‘Ÿ+1)2(𝑝+2)‖≀𝑑𝑣‖2(𝑝+2)2(𝑝+2)+𝐻(𝑑),βˆ€π‘‘β‰₯0.(3.36) Also, since (𝑋+π‘Œ)𝑠≀𝐢(𝑋𝑠+π‘Œπ‘ ),𝑋,π‘Œβ‰₯0,𝑠>0,(3.37) by using (3.24) and (3.34), we conclude that ‖𝑣‖2π›Ώπ‘š(𝑝+2)2(𝑝+2)β€–π‘’β€–π‘š+1π‘š+1‖≀𝐢𝑣‖2(𝑝+2)2(𝑝+2)+‖𝑒‖2(𝑝+2)(π‘š+1)‖≀𝐢𝑣‖2(𝑝+2)2(𝑝+2)+‖𝑒‖2(𝑝+2)2(𝑝+2),‖𝑒‖2π›Ώπ‘Ÿ(𝑝+2)2(𝑝+2)β€–π‘£β€–π‘Ÿ+1π‘Ÿ+1≀𝐢‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)(π‘Ÿ+1)≀𝐢‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2),(3.38) where 𝐢 is a generic positive constant. Taking into account (3.33)–(3.38), estimate (3.32) takes the form Ξ˜ξ…ž(𝑑)β‰₯((1βˆ’π›Ώ)βˆ’π‘€πœ–)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+πœ–2(π‘ž+1)βˆ’πΆ1𝑀1βˆ’π‘šβˆ’πΆ1𝑀2βˆ’π‘Ÿξ€Έξ€·π‘π»(𝑑)+πœ–2βˆ’πΆ2𝑀1βˆ’π‘šβˆ’πΆ2𝑀2βˆ’π‘Ÿξ€Έξ‚€β€–π‘’β€–2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2),(3.39) where 𝐢1=max{𝑐3𝑑+𝐢,𝑐4𝑑+𝐢}, 𝐢2=max{𝑐3𝑑,𝑐4𝑑}. At this point, and for large values of 𝑀1 and 𝑀2, we can find positive constants πœ…1 and πœ…2 such that (3.39) becomes Ξ˜ξ…ž(𝑑)β‰₯((1βˆ’π›Ώ)βˆ’π‘€πœ–)π»βˆ’π›Ώ(𝑑)π»ξ…žξ‚€β€–β€–π‘’(𝑑)+πœ–(π‘ž+2)𝑑‖‖22+‖‖𝑣𝑑‖‖22+πœ–πœ…1𝐻(𝑑)+πœ–πœ…2‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.40) Once 𝑀1 and 𝑀2 are fixed, we pick πœ– small enough so that (1βˆ’π›Ώ)βˆ’π‘€πœ–β‰₯0 and Θ(0)=𝐻1βˆ’π›Ώ(ξ€œ0)+πœ–Ξ©ξ€·π‘’0𝑒1+𝑣0𝑣1𝑑π‘₯>0.(3.41) Since π»ξ…ž(𝑑)β‰₯0, there exists Ξ›>0 such that (3.40) becomes Ξ˜ξ…žξ‚€β€–β€–π‘’(𝑑)β‰₯πœ–Ξ›π»(𝑑)+𝑑‖‖22+‖‖𝑣𝑑‖‖22+‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.42) Then, we have Θ(𝑑)β‰₯Θ(0),βˆ€π‘‘β‰₯0.(3.43)
Next, we have by HΓΆlder's and Young's inequalities ξ‚΅ξ€œΞ©π‘’π‘’π‘‘ξ€œπ‘‘π‘₯+Ω𝑣𝑣𝑑𝑑π‘₯1/(1βˆ’π›Ώ)ξ‚€β€–β‰€πΆπ‘’β€–πœ/(1βˆ’π›Ώ)2(𝑝+2)+‖‖𝑒𝑑‖‖2𝑠/(1βˆ’π›Ώ)+β€–π‘£β€–πœ/(1βˆ’π›Ώ)2(𝑝+2)+‖‖𝑣𝑑‖‖2𝑠/(1βˆ’π›Ώ),(3.44) for 1/𝜏+1/𝑠=1. We take 𝑠=2(1βˆ’π›Ώ), to get 𝜏/(1βˆ’π›Ώ)=2/(1βˆ’2𝛿). Here and in the sequel, 𝐢 denotes a positive constant which may change from line to line. By using (3.24) and (3.34), we have ‖𝑒‖2/(1βˆ’2𝛿)2(𝑝+2)‖≀𝑑𝑒‖2(𝑝+2)2(𝑝+2)+𝐻(𝑑),‖𝑣‖2/(1βˆ’2𝛿)2(𝑝+2)‖≀𝑑𝑣‖2(𝑝+2)2(𝑝+2)+𝐻(𝑑),βˆ€π‘‘β‰₯0.(3.45) Therefore, (3.44) becomes ξ‚΅ξ€œΞ©π‘’π‘’π‘‘ξ€œπ‘‘π‘₯+Ω𝑣𝑣𝑑𝑑π‘₯1/(1βˆ’π›Ώ)‖≀𝐢𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2)+‖‖𝑒𝑑‖‖22+‖‖𝑣𝑑‖‖22.(3.46) Note that Θ1/(1βˆ’π›Ώ)𝐻(𝑑)=1βˆ’π›Ώξ€œ(𝑑)+πœ–Ξ©ξ€·π‘’π‘’π‘‘+𝑣𝑣𝑑𝑑π‘₯1/(1βˆ’π›Ώ)||||ξ€œβ‰€πΆπ»(𝑑)+Ξ©π‘’π‘’π‘‘ξ€œπ‘‘π‘₯+Ω𝑣𝑣𝑑||||𝑑π‘₯1/(1βˆ’π›Ώ)ξƒͺ≀𝐢𝐻(𝑑)+‖𝑒‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2)+‖‖𝑒𝑑‖‖22+‖‖𝑣𝑑‖‖22.(3.47) Combining (3.42) with (3.47), we have Θ(𝑑)β‰₯𝐢Θ1/(1βˆ’π›Ώ)(𝑑),βˆ€π‘‘β‰₯0.(3.48) A simple application of Lemma 3.1 gives the desired result.


The authors are indebted to the referee for giving some important suggestions which improved the presentations of this paper. This work is supported in part by a China NSF Grant no. 10871097, Qing Lan Project of Jiangsu Province, the Foundation for Young Talents in College of Anhui Province Grant no. 2011SQRL115 and Program sponsored for scientific innovation research of college graduate in Jangsu province.


  1. A. Haraux and E. Zuazua, β€œDecay estimates for some semilinear damped hyperbolic problems,” Archive for Rational Mechanics and Analysis, vol. 100, no. 2, pp. 191–206, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. J. M. Ball, β€œRemarks on blow-up and nonexistence theorems for nonlinear evolution equations,” The Quarterly Journal of Mathematics. Oxford. Second Series, vol. 28, no. 112, pp. 473–486, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. V. K. Kalantarov, β€œThe occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,” Journal of Soviet Mathematics, vol. 10, no. 1, pp. 53–70, 1978. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. H. A. Levine, β€œInstability and nonexistence of global solutions to nonlinear wave equations of the form Putt=βˆ’Au+β„±(u),” Transactions of the American Mathematical Society, vol. 192, pp. 1–21, 1974. View at: Google Scholar | Zentralblatt MATH
  5. H. A. Levine, β€œSome additional remarks on the nonexistence of global solutions to nonlinear wave equations,” SIAM Journal on Mathematical Analysis, vol. 5, pp. 138–146, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. V. Georgiev and G. Todorova, β€œExistence of a solution of the wave equation with nonlinear damping and source terms,” Journal of Differential Equations, vol. 109, no. 2, pp. 295–308, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. S. A. Messaoudi, β€œBlow up in a nonlinearly damped wave equation,” Mathematische Nachrichten, vol. 231, pp. 105–111, 2001. View at: Google Scholar | Zentralblatt MATH
  8. S. A. Messaoudi, β€œBlow-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
  9. G. Andrews, β€œOn the existence of solutions to the equation utt=uxxt+Οƒ(ux)x,” Journal of Differential Equations, vol. 35, no. 2, pp. 200–231, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. G. Andrews and J. M. Ball, β€œAsymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity,” Journal of Differential Equations, vol. 44, no. 2, pp. 306–341, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. J. Clements, β€œExistence theorems for a quasilinear evolution equation,” SIAM Journal on Applied Mathematics, vol. 26, pp. 745–752, 1974. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. S. Kawashima and Y. Shibata, β€œGlobal existence and exponential stability of small solutions to nonlinear viscoelasticity,” Communications in Mathematical Physics, vol. 148, no. 1, pp. 189–208, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  13. D. D Ang and A. P. N. Dinh, β€œStrong solutions of a quasilinear wave equation with nonlinear damping,” SIAM Journal on Mathematical Analysis, vol. 19, no. 2, pp. 337–347, 1988. View at: Publisher Site | Google Scholar
  14. W. N. Findley, J. S. Lai, and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials, North-Holland, Amsterdam, The Netherlands, 1976.
  15. J. K. Knowles, β€œOn finite anti-plane shear for incompressible elastic materials,” Australian Mathematical Society. Journal. Series B, vol. 19, no. 4, pp. 400–415, 1975/76. View at: Google Scholar
  16. V. P. Maslov and P. P. Mosolov, Nonlinear Wave Equations Perturbed by Viscous Terms, vol. 31 of de Gruyter Expositions in Mathematics, Walter De Gruyter, Berlin, Germany, 2000.
  17. J. M. Greenberg, R. C. MacCamy, and V. J. Mizel, β€œOn the existence, uniqueness, and stability of solutions of the equation Οƒβ€²(ux)uxx+Ξ»uxxt=ρ0utt,” vol. 17, pp. 707–728, 1967/1968. View at: Google Scholar
  18. Y. Yamada, β€œSome remarks on the equation Yttβˆ’Οƒ(Yx)Yxxβˆ’Yxxt=f,” Osaka Journal of Mathematics, vol. 17, no. 2, pp. 303–323, 1980. View at: Google Scholar | Zentralblatt MATH
  19. Z. Yang and G. Chen, β€œGlobal existence of solutions for quasi-linear wave equations with viscous damping,” Journal of Mathematical Analysis and Applications, vol. 285, no. 2, pp. 604–618, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  20. G. Chen, H. Yue, and S. Wang, β€œThe initial boundary value problem for quasi-linear wave equation with viscous damping,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 823–839, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. J. Hao, Y. Zhang, and S. Li, β€œGlobal existence and blow-up phenomena for a nonlinear wave equation,” Nonlinear Analysis, vol. 71, no. 10, pp. 4823–4832, 2009. View at: Publisher Site | Google Scholar
  22. S. A. Messaoudi and B. Said Houari, β€œGlobal non-existence of solutions of a class of wave equations with non-linear damping and source terms,” Mathematical Methods in the Applied Sciences, vol. 27, no. 14, pp. 1687–1696, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. I. E. Segal, β€œNonlinear partial differential equations in quantum field theory,” in Proc. Sympos. Appl. Math., Vol. XVII, pp. 210–226, American Mathematical Society, Providence, RI, USA, 1965. View at: Google Scholar | Zentralblatt MATH
  24. K. Jorgens, Nonlinear Wave Equations, University of Colorado, Department of Mathematics, 1970.
  25. K. Agre and M. A. Rammaha, β€œSystems 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
  26. B. Said-Houari, β€œGlobal 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
  27. J. Wu, S. Li, and S. Chai, β€œExistence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source,” Nonlinear Analysis, vol. 72, no. 11, pp. 3969–3975, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  28. L. E. Payne and D. H. Sattinger, β€œSaddle points and instability of nonlinear hyperbolic equations,” Israel Journal of Mathematics, vol. 22, no. 3-4, pp. 273–303, 1975. View at: Publisher Site | Google Scholar
  29. E. Vitillaro, β€œGlobal 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
  30. S. A. Messaoudi and B. Said-Houari, β€œGlobal 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

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