Abstract

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 š‘¢š‘”š‘”ī‚€š‘”ī‚€||||āˆ’divāˆ‡š‘¢2ī‚ī‚+||š‘¢āˆ‡š‘¢š‘”||š‘šāˆ’1š‘¢š‘”=š‘“1š‘£(š‘¢,š‘£),(š‘„,š‘”)āˆˆĪ©Ć—(0,š‘‡),š‘”š‘”ī‚€š‘”ī‚€||||āˆ’divāˆ‡š‘£2ī‚ī‚+||š‘£āˆ‡š‘£š‘”||š‘Ÿāˆ’1š‘£š‘”=š‘“2(š‘¢,š‘£),(š‘„,š‘”)āˆˆĪ©Ć—(0,š‘‡),š‘¢(š‘„,š‘”)=š‘£(š‘„,š‘”)=0,š‘„āˆˆšœ•Ī©Ć—(0,š‘‡),š‘¢(š‘„,0)=š‘¢0(š‘„),š‘¢š‘”(š‘„,0)=š‘¢1(š‘„),š‘„āˆˆĪ©,š‘£(š‘„,0)=š‘£0(š‘„),š‘£š‘”(š‘„,0)=š‘£1(š‘„),š‘„āˆˆĪ©,(1.1) where Ī© is a bounded domain in ā„š‘› with a smooth boundary šœ•Ī©, š‘š,š‘Ÿā‰„1, and š‘“š‘–(ā‹…,ā‹…)āˆ¶ā„2ā†’ā„ (š‘–=1,2) are given functions to be specified later. We assume that š‘” is a function which satisfies š‘”āˆˆš¶1,š‘”(š‘ )>0,š‘”(š‘ )+2š‘ š‘”ī…ž(š‘ )>0(1.2) for š‘ >0.

To motivate our work, let us recall some results regarding š‘”ā‰”1. The single-wave equation of the form š‘¢š‘”š‘”ī€·š‘¢āˆ’Ī”š‘¢+ā„Žš‘”ī€ø=š‘“(š‘¢),š‘„āˆˆĪ©,š‘”>0(1.3) in Ī©Ć—(0,āˆž) 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, (š‘“=0), it is well known that the damping term ā„Ž(š‘¢š‘”) 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 ā„Ž(š‘¢š‘”)=š‘Žš‘¢š‘” and a polynomial source term of the form š‘“(š‘¢)=š‘|š‘¢|š‘āˆ’2š‘¢. He showed that solutions with negative initial energy blow up in finite time. The main tool used in [4, 5] is the ā€œconcavity method.ā€ Georgiev and Todorova in [6] extended Levine's result to the nonlinear damping case ā„Ž(š‘¢š‘”)=š‘Ž|š‘¢š‘”|š‘šāˆ’2š‘¢š‘”. In their work, the authors considered problem (1.3) with š‘“(š‘¢)=š‘|š‘¢|š‘āˆ’2š‘¢ 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 š‘šā‰„š‘ā‰„2 and blow up in finite time if š‘>š‘šā‰„2 and the initial energy is sufficiently negative. This latter result has been pushed by Messaoudi [7] to the situation where the initial energy šø(0)<0 and has been improved by the same author in [8] to accommodate certain solutions with positive initial energy.

In the case of š‘” being a given nonlinear function, the following equation: š‘¢š‘”š‘”ī€·š‘¢āˆ’š‘”š‘„ī€øš‘„āˆ’š‘¢š‘„š‘„š‘”||š‘¢+š›æš‘”||š‘šāˆ’1š‘¢š‘”=šœ‡|š‘¢|š‘āˆ’1š‘¢,š‘„āˆˆ(0,1),š‘”>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 [9ā€“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 [14ā€“16] for physical origins and derivation of mathematical models of motions of viscoelastic media and only recall here that, in applications, the unknown š‘¢ naturally represents the displacement of the body relative to a fixed reference configuration. When š›æ=š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 š›æ|š‘¢š‘”|š‘šāˆ’1š‘¢š‘” and šœ‡|š‘¢|š‘āˆ’1š‘¢ by š‘”(š‘¢š‘”) and š‘“(š‘¢), 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 š‘¢š‘”š‘”ī‚€š‘”ī‚€||||āˆ’divāˆ‡š‘¢2ī‚ī‚ī€·š‘¢āˆ‡š‘¢+ā„Žš‘”ī€ø=š‘“(š‘¢),š‘„āˆˆĪ©,š‘”>0(1.5) with initial and Dirichlet boundary condition, where š‘” satisfies condition (1.2) and š‘”(š‘ )ā‰„š‘1+š‘2š‘ š‘ž,š‘žā‰„0.(1.6) The damping term has the form ā„Žī€·š‘¢š‘”ī€ø=š‘‘1š‘¢š‘”+š‘‘2||š‘¢š‘”||š‘Ÿāˆ’1š‘¢š‘”,š‘Ÿ>1.(1.7) The source term is š‘“(š‘¢)=š‘Ž1š‘¢+š‘Ž2|š‘¢|š‘āˆ’1š‘¢(1.8) with š‘ā‰„1 for š‘›=1,2 and 1ā‰¤š‘›ā‰¤2š‘›/(š‘›āˆ’2) for š‘›ā‰„3, š‘Ž1,š‘Ž2,š‘1,š‘2,š‘‘1,š‘‘2 are nonnegative constants, and š‘1+š‘2>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 š‘”ā‰”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 š‘“1(š‘¢,š‘£) and š‘“2(š‘¢,š‘£) 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 šø(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 ||ā‹…||š‘ž the šæš‘ž(Ī©) norm for 1ā‰¤š‘žā‰¤āˆž and by ||āˆ‡ā‹…||2 the Dirichlet norm in š»10(Ī©) which is equivalent to the š»1(Ī©) norm. Moreover, we set (ī€œšœ‘,šœ“)=Ī©šœ‘(š‘„)šœ“(š‘„)š‘‘š‘„(2.1) as the usual šæ2(Ī©) inner product.

Concerning the functions š‘“1(š‘¢,š‘£) and š‘“2(š‘¢,š‘£), we take š‘“1ī€ŗ(š‘¢,š‘£)=š‘Ž|š‘¢+š‘£|2(š‘+1)(š‘¢+š‘£)+š‘|š‘¢|š‘š‘¢|š‘£|(š‘+2)ī€»,š‘“2ī€ŗš‘Ž(š‘¢,š‘£)=|š‘¢+š‘£|2(š‘+1)(š‘¢+š‘£)+š‘|š‘¢|(š‘+2)|š‘£|š‘š‘£ī€»,(2.2) where š‘Ž,š‘>0 are constants and š‘ satisfies āŽ§āŽŖāŽØāŽŖāŽ©š‘>āˆ’1,ifš‘›=1,2,āˆ’1<š‘ā‰¤4āˆ’š‘›š‘›āˆ’2,ifš‘›ā‰„3.(2.3) One can easily verify that š‘¢š‘“1(š‘¢,š‘£)+š‘£š‘“2(š‘¢,š‘£)=2(š‘+2)š¹(š‘¢,š‘£),āˆ€(š‘¢,š‘£)āˆˆā„2,(2.4) where 1š¹(š‘¢,š‘£)=2ī€ŗ(š‘+2)š‘Ž|š‘¢+š‘£|2(š‘+2)+2š‘|š‘¢š‘£|š‘+2ī€».(2.5)

We have the following result.

Lemma 2.1 (see [30, Lemmaā€‰ā€‰2.1]). There exist two positive constants š‘0 and š‘1 such that š‘0ī€·2(š‘+2)|š‘¢|2(š‘+2)+|š‘£|2(š‘+2)ī€øš‘ā‰¤š¹(š‘¢,š‘£)ā‰¤1ī€·2(š‘+2)|š‘¢|2(š‘+2)+|š‘£|2(š‘+2)ī€ø.(2.6)

Throughout this paper, we define š‘” by š‘”(š‘ )=š‘1+š‘2š‘ š‘ž,š‘žā‰„0,š‘1+š‘2>0,(2.7) where š‘1,š‘2 are nonnegative constants. Obviously, š‘” satisfies conditions (1.2) and (1.6). Set ī€œšŗ(š‘ )=š‘ 0š‘”(š‘ )š‘‘š‘ ,š‘ ā‰„0.(2.8)

In order to state and prove our result, we introduce the following function space: ī‚†š‘=(š‘¢,š‘£)āˆ£š‘¢,š‘£āˆˆšæāˆžī‚€[0,š‘‡);š‘Š01,2(š‘ž+1)(Ī©)āˆ©šæ2(š‘+2)ī‚,š‘¢(Ī©)š‘”āˆˆšæāˆžī€·[0,š‘‡);šæ2ī€ø(Ī©)āˆ©šæš‘š+1š‘£(Ī©Ć—(0,š‘‡)),š‘”āˆˆšæāˆžī€·[0,š‘‡);šæ2ī€ø(Ī©)āˆ©šæš‘Ÿ+1(Ī©Ć—(0,š‘‡)),š‘¢š‘”š‘”,š‘£š‘”š‘”āˆˆšæāˆžī€·[0,š‘‡),šæ2ī€øī‚‡.(Ī©)(2.9)

Define the energy functional šø(š‘”) associated with our system 1šø(š‘”)=2ī‚€ā€–ā€–š‘¢š‘”(ā€–ā€–š‘”)22+ā€–ā€–š‘£š‘”(ā€–ā€–š‘”)22ī‚+12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī€œī‚ī‚š‘‘š‘„āˆ’Ī©š¹(š‘¢,š‘£)š‘‘š‘„.(2.10) A simple computation gives š‘‘šø(š‘”)š‘‘š‘”=āˆ’ā€–š‘¢ā€–š‘š+1š‘š+1āˆ’ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1ā‰¤0.(2.11)

Our main result reads as follows.

Theorem 2.2. Assume that (2.3) holds. Assume further that 2(š‘+2)>max{2š‘ž+2,š‘š+1,š‘Ÿ+1}. Then any solution of (1.1) with initial data satisfying ī‚µī€œĪ©ī‚€šŗī‚€||āˆ‡š‘¢0||2ī‚ī‚€||+šŗāˆ‡š‘£0||2ī‚¶ī‚ī‚š‘‘š‘„1/2>š›¼1,šø(0)<šø2,(2.12) cannot exist for all time, where the constant š›¼1 and šø2 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 Ī˜(š‘”) be a solution of the ordinary differential inequality š‘‘Ī˜(š‘”)š‘‘š‘”ā‰„š¶Ī˜1+šœ€(š‘”),š‘”>0,(3.1) where šœ€>0. If Ī˜(0)>0, then the solution ceases to exist for š‘”ā‰„Ī˜āˆ’šœ€(0)š¶āˆ’1šœ€āˆ’1.

Lemma 3.2. Assume that (2.3) holds. Then there exists šœ‚>0 such that for any (š‘¢,š‘£)āˆˆš‘, one has ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚µī€œā‰¤šœ‚Ī©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī‚¶ī‚ī‚š‘‘š‘„š‘+2.(3.2)

Proof. By using Minkowski's inequality, we get ā€–š‘¢+š‘£ā€–22(š‘+2)ī‚€ā€–ā‰¤2š‘¢ā€–22(š‘+2)+ā€–š‘£ā€–22(š‘+2)ī‚.(3.3) Also, Hƶlder's and Young's inequalities give us ā€–š‘¢š‘£ā€–š‘+2ā‰¤ā€–š‘¢ā€–2(š‘+2)ā€–š‘£ā€–2(š‘+2)ā‰¤12ī‚€ā€–š‘¢ā€–22(š‘+2)+ā€–š‘£ā€–22(š‘+2)ī‚.(3.4) If š‘1>0, then we have ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī€·ī‚ī‚š‘‘š‘„ā‰„š‘ā€–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€ø.(3.5) If š‘1=0, from š‘1+š‘2>0, we have š‘2>0. Since š‘Š01,2(š‘ž+1)(Ī©)ā†Ŗš»10(Ī©), we have ā€–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ā‰¤š‘1ī‚€ā€–āˆ‡š‘¢ā€–22(š‘ž+1)+ā€–āˆ‡š‘£ā€–22(š‘ž+1)ī‚,(3.6) which implies that (3.5) still holds for š‘1=0. Combining (3.3), (3.4) with (3.5) and the embedding š»10(Ī©)ā†Ŗšæ2(š‘+2)(Ī©), we have (3.2).

In order to prove our result and for the sake of simplicity, we take š‘Ž=š‘=1 and introduce the following: šµ=šœ‚1/(2(š‘+2)),š›¼1=šµāˆ’(š‘+2/(š‘+1)),šø1=ī‚µ12āˆ’1ī‚¶š›¼2(š‘+2)21,šø2=ī‚µ1āˆ’12(š‘ž+1)ī‚¶š›¼2(š‘+2)21,(3.7) where š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 (š‘¢,š‘£)āˆˆš‘ be the solution of the system (1.1). Assume further that šø(0)<šø1 and ī‚µī€œĪ©ī‚€šŗī‚€||āˆ‡š‘¢0||2ī‚ī‚€||+šŗāˆ‡š‘£0||2ī‚¶ī‚ī‚š‘‘š‘„1/2>š›¼1.(3.8) Then there exists a constant š›¼2>š›¼1 such that ī‚µī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī‚¶ī‚ī‚š‘‘š‘„1/2ā‰„š›¼2ī‚€,forš‘”>0,(3.9)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚1/(2(š‘+2))ā‰„šµš›¼2,forš‘”>0.(3.10)

Proof. We first note that, by (2.10), (3.2), and the definition of šµ, we have 1šø(š‘”)ā‰„2ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£21ī‚ī‚š‘‘š‘„āˆ’ī‚€2(š‘+2)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚ā‰„12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2šµī‚ī‚š‘‘š‘„āˆ’2(š‘+2)ī‚µī€œ2(š‘+2)Ī©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī‚¶ī‚ī‚š‘‘š‘„š‘+2=12š›¼2āˆ’šµ2(š‘+2)š›¼2(š‘+2)2(š‘+2),(3.11) where āˆ«š›¼=(Ī©(šŗ(|āˆ‡š‘¢|2)+šŗ(|āˆ‡š‘£|2))š‘‘š‘„)1/2. It is not hard to verify that š‘” is increasing for 0<š›¼<š›¼1, decreasing for š›¼>š›¼1, š‘”(š›¼)ā†’āˆ’āˆž as š›¼ā†’+āˆž, and š‘”ī€·š›¼1ī€ø=12š›¼21āˆ’šµ2(š‘+2)š›¼2(š‘+2)12(š‘+2)=šø1,(3.12) where š›¼1 is given in (3.7). Since šø(0)<šø1, there exists š›¼2>š›¼1 such that š‘”(š›¼2)=šø(0).
Set š›¼0āˆ«=(Ī©(šŗ(|āˆ‡š‘¢0|2)+šŗ(|āˆ‡š‘£0|2))š‘‘š‘„)1/2. Then by (3.11) we get š‘”(š›¼0)ā‰¤šø(0)=š‘”(š›¼2), which implies that š›¼0ā‰„š›¼2. Now, to establish (3.9), we suppose by contradiction that ī‚µī€œĪ©ī‚€šŗī‚€||ī€·š‘”āˆ‡š‘¢0ī€ø||2ī‚ī‚€||ī€·š‘”+šŗāˆ‡š‘£0ī€ø||2ī‚¶ī‚ī‚š‘‘š‘„1/2<š›¼2,(3.13) for some š‘”0>0. By the continuity of āˆ«Ī©(šŗ(|āˆ‡š‘¢|2)+šŗ(|āˆ‡š‘£|2))š‘‘š‘„, we can choose š‘”0 such that ī‚µī€œĪ©ī‚€šŗī‚€||ī€·š‘”āˆ‡š‘¢0ī€ø||2ī‚ī‚€||ī€·š‘”+šŗāˆ‡š‘£0ī€ø||2ī‚¶ī‚ī‚š‘‘š‘„1/2>š›¼1.(3.14) Again, the use of (3.11) leads to šøī€·š‘”0ī€øīƒ©ī‚µī€œā‰„š‘”Ī©ī‚€šŗī‚€||ī€·š‘”āˆ‡š‘¢0ī€ø||2ī‚ī‚€||ī€·š‘”+šŗāˆ‡š‘£0ī€ø||2ī‚¶ī‚ī‚š‘‘š‘„1/2īƒŖī€·š›¼>š‘”2ī€ø=šø(0).(3.15) This is impossible since šø(š‘”)ā‰¤šø(0) for all š‘”āˆˆ[0,š‘‡). Hence (3.9) is established.
To prove (3.10), we make use of (2.10) to get 12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£21ī‚ī‚š‘‘š‘„ā‰¤šø(0)+ī‚€2(š‘+2)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚.(3.16) Consequently, (3.9) yields 1ī‚€2(š‘+2)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚ā‰„12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ā‰„1ī‚ī‚š‘‘š‘„āˆ’šø(0)2š›¼221āˆ’šø(0)ā‰„2š›¼22ī€·š›¼āˆ’š‘”2ī€ø=šµ2(š‘+2)2š›¼(š‘+2)22(š‘+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 š»(š‘”)=šø2āˆ’šø(š‘”).(3.18) By using (2.10) and (3.18), we have 0<š»(0)ā‰¤š»(š‘”)=šø2āˆ’12ī‚€ā€–ā€–š‘¢š‘”(ā€–ā€–š‘”)22+ā€–ā€–š‘£š‘”(ā€–ā€–š‘”)22ī‚āˆ’12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2+1ī‚ī‚š‘‘š‘„ī‚€2(š‘+2)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚.(3.19) From (3.9), we have šø2āˆ’12ī‚€ā€–ā€–š‘¢š‘”(ā€–ā€–š‘”)22+ā€–ā€–š‘£š‘”(ā€–ā€–š‘”)22ī‚āˆ’12ī€œĪ©ī‚€šŗī‚€||||āˆ‡š‘¢2ī‚ī‚€||||+šŗāˆ‡š‘£2ī‚ī‚š‘‘š‘„ā‰¤šø2āˆ’12š›¼21ā‰¤šø1āˆ’12š›¼211=āˆ’š›¼2(š‘+2)21<0,āˆ€š‘”ā‰„0.(3.20) Hence, by the above inequality and (2.6), we have 10<š»(0)ā‰¤š»(š‘”)ā‰¤2ī‚€(š‘+2)ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚,ā‰¤š‘(3.21)1ī‚€2(š‘+2)ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚.(3.22) We then define Ī˜(š‘”)=š»1āˆ’š›æ(ī€œš‘”)+šœ–Ī©ī€·š‘¢š‘¢š‘”+š‘£š‘£š‘”ī€øš‘‘š‘„,(3.23) where šœ– small enough is to be chosen later and ī‚»0<š›æā‰¤minš‘+1,2(š‘+2)2(š‘+2)āˆ’(š‘š+1),2š‘š(š‘+2)2(š‘+2)āˆ’(š‘Ÿ+1)ī‚¼2š‘Ÿ(š‘+2).(3.24) Our goal is to show that Ī˜(š‘”) satisfies the differential inequality (3.1) which leads to a blow-up in finite time. By taking a derivative of (3.23), we get Ī˜ī…ž(š‘”)=(1āˆ’š›æ)š»āˆ’š›æ(š‘”)š»ī…ž(ī‚€ā€–ā€–š‘¢š‘”)+šœ–š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚ī€œāˆ’šœ–Ī©ī‚€š‘”ī‚€||||āˆ‡š‘¢2ī‚||||āˆ‡š‘¢2ī‚€||||+š‘”āˆ‡š‘£2ī‚||||āˆ‡š‘£2ī‚ī€œš‘‘š‘„āˆ’šœ–Ī©ī‚€||š‘¢š‘”||š‘šāˆ’1š‘¢š‘”||š‘£š‘¢+š‘”||š‘Ÿāˆ’1š‘£š‘”š‘£ī‚ī€œš‘‘š‘„+šœ–Ī©ī€·š‘¢š‘“1(š‘¢,š‘£)+š‘£š‘“2ī€ø(š‘¢,š‘£)š‘‘š‘„=(1āˆ’š›æ)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚āˆ’š‘1šœ–ī€·ā€–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€øāˆ’šœ–š‘2ā€–āˆ‡š‘¢ā€–2(š‘ž+2)2(š‘ž+2)āˆ’šœ–š‘2ā€–āˆ‡š‘£ā€–2(š‘ž+2)2(š‘ž+2)ī€œāˆ’šœ–Ī©ī‚€||š‘¢š‘”||š‘šāˆ’1š‘¢š‘”||š‘£š‘¢+š‘”||š‘Ÿāˆ’1š‘£š‘”š‘£ī‚ī‚€š‘‘š‘„+šœ–ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚.(3.25) From the definition of š»(š‘”), it follows that āˆ’š‘2ā€–āˆ‡š‘¢ā€–2(š‘ž+2)2(š‘ž+2)āˆ’š‘2ā€–āˆ‡š‘£ā€–2(š‘ž+2)2(š‘ž+2)=2(š‘ž+1)š»(š‘”)āˆ’2(š‘ž+1)šø2ī‚€ā€–ā€–š‘¢+(š‘ž+1)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+(š‘ž+1)š‘1ī€·ā€–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€øī€œāˆ’2(š‘ž+1)Ī©š¹(š‘¢,š‘£)š‘‘š‘„,(3.26) which together with (3.25) gives Ī˜ā€²(š‘”)=(1āˆ’š›æ)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+š‘1ī€·ā€–š‘žšœ–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€øī€œāˆ’šœ–Ī©ī‚€||š‘¢š‘”||š‘šāˆ’1š‘¢š‘”||š‘£š‘¢+š‘”||š‘Ÿāˆ’1š‘£š‘”š‘£ī‚ī‚µš‘‘š‘„+šœ–1āˆ’š‘ž+1ī‚¶ī‚€ā€–š‘+2š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚+2(š‘ž+1)š»(š‘”)āˆ’2(š‘ž+1)šø2.(3.27) Then, using (3.10), we obtain Ī˜ā€²(š‘”)ā‰„(1āˆ’š›æ)š»āˆ’š›æī‚€ā€–ā€–š‘¢(š‘”)š»ā€²(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+š‘1ī€·ā€–š‘žšœ–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€ø+2(š‘ž+1)š»(š‘”)+šœ–š‘ī‚€ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚ī€œāˆ’šœ–Ī©ī‚€||š‘¢š‘”||š‘šāˆ’1š‘¢š‘”||š‘£š‘¢+š‘”||š‘Ÿāˆ’1š‘£š‘”š‘£ī‚š‘‘š‘„,(3.28) where š‘=1āˆ’(š‘ž+1)/(š‘+2)āˆ’2(š‘ž+1)šø2(šµš›¼2)āˆ’2(š‘+2). It is clear that š‘>0, since š›¼2>šµāˆ’(š‘+2)/(š‘+1). We now exploit Young's inequality to estimate the last two terms on the right side of (3.28) ||||ī€œĪ©||š‘¢š‘”||š‘šāˆ’1š‘¢š‘”||||ā‰¤šœ‚š‘¢š‘‘š‘„1š‘š+1š‘š+1ā€–š‘¢ā€–š‘š+1š‘š+1+š‘ššœ‚1āˆ’((š‘š+1)/š‘š)ā€–ā€–š‘¢š‘š+1š‘”ā€–ā€–š‘š+1š‘š+1,||||ī€œĪ©||š‘£š‘”||š‘Ÿāˆ’1š‘£š‘”||||ā‰¤šœ‚š‘£š‘‘š‘„2š‘Ÿ+1š‘Ÿ+1ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1+š‘Ÿšœ‚2āˆ’((š‘Ÿ+1)/š‘Ÿ)ā€–ā€–š‘£š‘Ÿ+1š‘”ā€–ā€–š‘Ÿ+1š‘Ÿ+1,(3.29) where šœ‚1,šœ‚2 are parameters depending on the time š‘” and specified later. Inserting the last two estimates into (3.28), we have Ī˜ī…ž(š‘”)ā‰„(1āˆ’š›æ)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+š‘1ī€·ā€–š‘žšœ–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€ø+2(š‘ž+1)š»(š‘”)+šœ–š‘ī‚€ā€–š‘¢+š‘£ā€–2(š‘+2)2(š‘+2)+2ā€–š‘¢š‘£ā€–š‘+2š‘+2ī‚šœ‚āˆ’šœ–1š‘š+1š‘š+1ā€–š‘¢ā€–š‘š+1š‘š+1āˆ’šœ–š‘ššœ‚1āˆ’((š‘š+1)/š‘š)ā€–ā€–š‘¢š‘š+1š‘”ā€–ā€–š‘š+1š‘š+1šœ‚āˆ’šœ–2š‘Ÿ+1ā€–š‘Ÿ+1š‘£ā€–š‘Ÿ+1š‘Ÿ+1āˆ’šœ–š‘Ÿšœ‚2āˆ’((š‘Ÿ+1)/š‘Ÿ)ā€–ā€–š‘£š‘Ÿ+1š‘”ā€–ā€–š‘Ÿ+1š‘Ÿ+1.(3.30) By choosing šœ‚1 and šœ‚2 such that šœ‚1āˆ’(š‘š+1)/š‘š=š‘€1š»āˆ’š›æ(š‘”),šœ‚2āˆ’(š‘Ÿ+1)/š‘Ÿ=š‘€2š»āˆ’š›æ(š‘”),(3.31) where š‘€1 and š‘€2 are constants to be fixed later. Thus, by using (2.6) and (3.31), inequality (3.31) then takes the form Ī˜ī…ž(š‘”)ā‰„((1āˆ’š›æ)āˆ’š‘€šœ–)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+š‘1ī€·ā€–š‘žšœ–āˆ‡š‘¢ā€–22+ā€–āˆ‡š‘£ā€–22ī€ø+2(š‘ž+1)š»(š‘”)+šœ–š‘2ī‚€ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+2ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚āˆ’šœ–š‘€1āˆ’š‘šš»š›æš‘š(š‘”)ā€–š‘¢ā€–š‘š+1š‘š+1āˆ’šœ–š‘€2āˆ’š‘Ÿš»š›æš‘Ÿ(š‘”)ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1,(3.32) where š‘€=š‘š/(š‘š+1)š‘€1+š‘Ÿ/(š‘Ÿ+1)š‘€2 and š‘2 is a positive constant.
Since 2(š‘+2)>max{š‘š+1,š‘Ÿ+1}, taking into account (2.6) and (3.21), then we have š»š›æš‘šā€–(š‘”)š‘¢ā€–š‘š+1š‘š+1ā‰¤š‘3ī‚€ā€–š‘¢ā€–2š›æš‘š(š‘+2)+(š‘š+1)2(š‘+2)+ā€–š‘£ā€–2š›æš‘š(š‘+2)2(š‘+2)ā€–š‘¢ā€–š‘š+1š‘š+1ī‚,š»š›æš‘Ÿ(š‘”)ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1ā‰¤š‘4ī‚€ā€–š‘£ā€–2š›æš‘Ÿ(š‘+2)+(š‘Ÿ+1)2(š‘+2)+ā€–š‘¢ā€–2š›æš‘Ÿ(š‘+2)2(š‘+2)ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1ī‚,(3.33) for some positive constants š‘3 and š‘4. By using (3.24) and the algebraic inequality š‘§šœˆī‚€1ā‰¤š‘§+1ā‰¤1+š‘Žī‚(š‘§+š‘Ž),āˆ€š‘§ā‰„0,0<šœˆā‰¤1,š‘Žā‰„0,(3.34) we have ā€–š‘¢ā€–2š›æš‘š(š‘+2)+(š‘š+1)2(š‘+2)ī‚€ā€–ā‰¤š‘‘š‘¢ā€–2(š‘+2)2(š‘+2)ī‚ī‚€ā€–+š»(0)ā‰¤š‘‘š‘¢ā€–2(š‘+2)2(š‘+2)ī‚+š»(š‘”),āˆ€š‘”ā‰„0,(3.35) where š‘‘=1+1/š»(0). Similarly, ā€–š‘£ā€–2š›æš‘Ÿ(š‘+2)+(š‘Ÿ+1)2(š‘+2)ī‚€ā€–ā‰¤š‘‘š‘£ā€–2(š‘+2)2(š‘+2)ī‚+š»(š‘”),āˆ€š‘”ā‰„0.(3.36) Also, since (š‘‹+š‘Œ)š‘ ā‰¤š¶(š‘‹š‘ +š‘Œš‘ ),š‘‹,š‘Œā‰„0,š‘ >0,(3.37) by using (3.24) and (3.34), we conclude that ā€–š‘£ā€–2š›æš‘š(š‘+2)2(š‘+2)ā€–š‘¢ā€–š‘š+1š‘š+1ī‚€ā€–ā‰¤š¶š‘£ā€–2(š‘+2)2(š‘+2)+ā€–š‘¢ā€–2(š‘+2)(š‘š+1)ī‚ī‚€ā€–ā‰¤š¶š‘£ā€–2(š‘+2)2(š‘+2)+ā€–š‘¢ā€–2(š‘+2)2(š‘+2)ī‚,ā€–š‘¢ā€–2š›æš‘Ÿ(š‘+2)2(š‘+2)ā€–š‘£ā€–š‘Ÿ+1š‘Ÿ+1ī‚€ā‰¤š¶ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)(š‘Ÿ+1)ī‚ī‚€ā‰¤š¶ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚,(3.38) where š¶ is a generic positive constant. Taking into account (3.33)ā€“(3.38), estimate (3.32) takes the form Ī˜ī…ž(š‘”)ā‰„((1āˆ’š›æ)āˆ’š‘€šœ–)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚ī€·+šœ–2(š‘ž+1)āˆ’š¶1š‘€1āˆ’š‘šāˆ’š¶1š‘€2āˆ’š‘Ÿī€øī€·š‘š»(š‘”)+šœ–2āˆ’š¶2š‘€1āˆ’š‘šāˆ’š¶2š‘€2āˆ’š‘Ÿī€øī‚€ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚,(3.39) where š¶1=max{š‘3š‘‘+š¶,š‘4š‘‘+š¶}, š¶2=max{š‘3š‘‘,š‘4š‘‘}. At this point, and for large values of š‘€1 and š‘€2, we can find positive constants šœ…1 and šœ…2 such that (3.39) becomes Ī˜ī…ž(š‘”)ā‰„((1āˆ’š›æ)āˆ’š‘€šœ–)š»āˆ’š›æ(š‘”)š»ī…žī‚€ā€–ā€–š‘¢(š‘”)+šœ–(š‘ž+2)š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚+šœ–šœ…1š»(š‘”)+šœ–šœ…2ī‚€ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚.(3.40) Once š‘€1 and š‘€2 are fixed, we pick šœ– small enough so that (1āˆ’š›æ)āˆ’š‘€šœ–ā‰„0 and Ī˜(0)=š»1āˆ’š›æ(ī€œ0)+šœ–Ī©ī€·š‘¢0š‘¢1+š‘£0š‘£1ī€øš‘‘š‘„>0.(3.41) Since š»ī…ž(š‘”)ā‰„0, there exists Ī›>0 such that (3.40) becomes Ī˜ī…žī‚€ā€–ā€–š‘¢(š‘”)ā‰„šœ–Ī›š»(š‘”)+š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22+ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)ī‚.(3.42) Then, we have Ī˜(š‘”)ā‰„Ī˜(0),āˆ€š‘”ā‰„0.(3.43)
Next, we have by Hƶlder's and Young's inequalities ī‚µī€œĪ©š‘¢š‘¢š‘”ī€œš‘‘š‘„+Ī©š‘£š‘£š‘”ī‚¶š‘‘š‘„1/(1āˆ’š›æ)ī‚€ā€–ā‰¤š¶š‘¢ā€–šœ/(1āˆ’š›æ)2(š‘+2)+ā€–ā€–š‘¢š‘”ā€–ā€–2š‘ /(1āˆ’š›æ)+ā€–š‘£ā€–šœ/(1āˆ’š›æ)2(š‘+2)+ā€–ā€–š‘£š‘”ā€–ā€–2š‘ /(1āˆ’š›æ)ī‚,(3.44) for 1/šœ+1/š‘ =1. We take š‘ =2(1āˆ’š›æ), to get šœ/(1āˆ’š›æ)=2/(1āˆ’2š›æ). Here and in the sequel, š¶ denotes a positive constant which may change from line to line. By using (3.24) and (3.34), we have ā€–š‘¢ā€–2/(1āˆ’2š›æ)2(š‘+2)ī‚€ā€–ā‰¤š‘‘š‘¢ā€–2(š‘+2)2(š‘+2)ī‚+š»(š‘”),ā€–š‘£ā€–2/(1āˆ’2š›æ)2(š‘+2)ī‚€ā€–ā‰¤š‘‘š‘£ā€–2(š‘+2)2(š‘+2)ī‚+š»(š‘”),āˆ€š‘”ā‰„0.(3.45) Therefore, (3.44) becomes ī‚µī€œĪ©š‘¢š‘¢š‘”ī€œš‘‘š‘„+Ī©š‘£š‘£š‘”ī‚¶š‘‘š‘„1/(1āˆ’š›æ)ī‚€ā€–ā‰¤š¶š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)+ā€–ā€–š‘¢š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚.(3.46) Note that Ī˜1/(1āˆ’š›æ)ī‚µš»(š‘”)=1āˆ’š›æī€œ(š‘”)+šœ–Ī©ī€·š‘¢š‘¢š‘”+š‘£š‘£š‘”ī€øī‚¶š‘‘š‘„1/(1āˆ’š›æ)īƒ©||||ī€œā‰¤š¶š»(š‘”)+Ī©š‘¢š‘¢š‘”ī€œš‘‘š‘„+Ī©š‘£š‘£š‘”||||š‘‘š‘„1/(1āˆ’š›æ)īƒŖī‚€ā‰¤š¶š»(š‘”)+ā€–š‘¢ā€–2(š‘+2)2(š‘+2)+ā€–š‘£ā€–2(š‘+2)2(š‘+2)+ā€–ā€–š‘¢š‘”ā€–ā€–22+ā€–ā€–š‘£š‘”ā€–ā€–22ī‚.(3.47) Combining (3.42) with (3.47), we have Ī˜(š‘”)ā‰„š¶Ī˜1/(1āˆ’š›æ)(š‘”),āˆ€š‘”ā‰„0.(3.48) A simple application of Lemma 3.1 gives the desired result.

Acknowledgments

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.