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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 760209, 14 pages
Research Article

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

1Jiangsu Provincial Key Laboratory for NSLSCS, School of Mathematical Science, Nanjing Normal University, Nanjing 210046, China
2Department of Mathematics, Anhui Science and Technology University, Fengyang 233100, Anhui, China

Received 26 November 2010; Revised 5 June 2011; Accepted 27 June 2011

Academic Editor: Josef Diblík

Copyright © 2011 Liang Fei and Gao Hongjun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 𝑢𝑡𝑡𝑔||||div𝑢2+||𝑢𝑢𝑡||𝑚1𝑢𝑡=𝑓1𝑣(𝑢,𝑣),(𝑥,𝑡)Ω×(0,𝑇),𝑡𝑡𝑔||||div𝑣2+||𝑣𝑣𝑡||𝑟1𝑣𝑡=𝑓2(𝑢,𝑣),(𝑥,𝑡)Ω×(0,𝑇),𝑢(𝑥,𝑡)=𝑣(𝑥,𝑡)=0,𝑥𝜕Ω×(0,𝑇),𝑢(𝑥,0)=𝑢0(𝑥),𝑢𝑡(𝑥,0)=𝑢1(𝑥),𝑥Ω,𝑣(𝑥,0)=𝑣0(𝑥),𝑣𝑡(𝑥,0)=𝑣1(𝑥),𝑥Ω,(1.1) where Ω is a bounded domain in 𝑛 with a smooth boundary 𝜕Ω, 𝑚,𝑟1, and 𝑓𝑖(,)2 (𝑖=1,2) are given functions to be specified later. We assume that 𝑔 is a function which satisfies 𝑔𝐶1,𝑔(𝑠)>0,𝑔(𝑠)+2𝑠𝑔(𝑠)>0(1.2) for 𝑠>0.

To motivate our work, let us recall some results regarding 𝑔1. The single-wave equation of the form 𝑢𝑡𝑡𝑢Δ𝑢+𝑡=𝑓(𝑢),𝑥Ω,𝑡>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 (𝑢𝑡) 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 (𝑢𝑡)=𝑎𝑢𝑡 and a polynomial source term of the form 𝑓(𝑢)=𝑏|𝑢|𝑝2𝑢. He showed that solutions with negative initial energy blow up in finite time. The main tool used in [4, 5] is the “concavity method.” Georgiev and Todorova in [6] extended Levine's result to the nonlinear damping case (𝑢𝑡)=𝑎|𝑢𝑡|𝑚2𝑢𝑡. In their work, the authors considered problem (1.3) with 𝑓(𝑢)=𝑏|𝑢|𝑝2𝑢 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 𝑚𝑝2 and blow up in finite time if 𝑝>𝑚2 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: 𝑢𝑡𝑡𝑢𝑔𝑥𝑥𝑢𝑥𝑥𝑡||𝑢+𝛿𝑡||𝑚1𝑢𝑡=𝜇|𝑢|𝑝1𝑢,𝑥(0,1),𝑡>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 [912]. 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 [1416] for physical origins and derivation of mathematical models of motions of viscoelastic media and only recall here that, in applications, the unknown 𝑢 naturally represents the displacement of the body relative to a fixed reference configuration. When 𝛿=𝜇=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|𝑢𝑡|𝑚1𝑢𝑡 and 𝜇|𝑢|𝑝1𝑢 by 𝑔(𝑢𝑡) and 𝑓(𝑢), 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 𝑢𝑡𝑡𝑔||||div𝑢2𝑢𝑢+𝑡=𝑓(𝑢),𝑥Ω,𝑡>0(1.5) with initial and Dirichlet boundary condition, where 𝑔 satisfies condition (1.2) and 𝑔(𝑠)𝑏1+𝑏2𝑠𝑞,𝑞0.(1.6) The damping term has the form 𝑢𝑡=𝑑1𝑢𝑡+𝑑2||𝑢𝑡||𝑟1𝑢𝑡,𝑟>1.(1.7) The source term is 𝑓(𝑢)=𝑎1𝑢+𝑎2|𝑢|𝑝1𝑢(1.8) with 𝑝1 for 𝑛=1,2 and 1𝑛2𝑛/(𝑛2) for 𝑛3, 𝑎1,𝑎2,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 𝑔1 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(𝑢,𝑣) and 𝑓2(𝑢,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 ||||𝑞 the 𝐿𝑞(Ω) norm for 1𝑞 and by ||||2 the Dirichlet norm in 𝐻10(Ω) which is equivalent to the 𝐻1(Ω) norm. Moreover, we set (𝜑,𝜓)=Ω𝜑(𝑥)𝜓(𝑥)𝑑𝑥(2.1) as the usual 𝐿2(Ω) inner product.

Concerning the functions 𝑓1(𝑢,𝑣) and 𝑓2(𝑢,𝑣), we take 𝑓1(𝑢,𝑣)=𝑎|𝑢+𝑣|2(𝑝+1)(𝑢+𝑣)+𝑏|𝑢|𝑝𝑢|𝑣|(𝑝+2),𝑓2𝑎(𝑢,𝑣)=|𝑢+𝑣|2(𝑝+1)(𝑢+𝑣)+𝑏|𝑢|(𝑝+2)|𝑣|𝑝𝑣,(2.2) where 𝑎,𝑏>0 are constants and 𝑝 satisfies 𝑝>1,if𝑛=1,2,1<𝑝4𝑛𝑛2,if𝑛3.(2.3) One can easily verify that 𝑢𝑓1(𝑢,𝑣)+𝑣𝑓2(𝑢,𝑣)=2(𝑝+2)𝐹(𝑢,𝑣),(𝑢,𝑣)2,(2.4) where 1𝐹(𝑢,𝑣)=2(𝑝+2)𝑎|𝑢+𝑣|2(𝑝+2)+2𝑏|𝑢𝑣|𝑝+2.(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 𝑐02(𝑝+2)|𝑢|2(𝑝+2)+|𝑣|2(𝑝+2)𝑐𝐹(𝑢,𝑣)12(𝑝+2)|𝑢|2(𝑝+2)+|𝑣|2(𝑝+2).(2.6)

Throughout this paper, we define 𝑔 by 𝑔(𝑠)=𝑏1+𝑏2𝑠𝑞,𝑞0,𝑏1+𝑏2>0,(2.7) where 𝑏1,𝑏2 are nonnegative constants. Obviously, 𝑔 satisfies conditions (1.2) and (1.6). Set 𝐺(𝑠)=𝑠0𝑔(𝑠)𝑑𝑠,𝑠0.(2.8)

In order to state and prove our result, we introduce the following function space: 𝑍=(𝑢,𝑣)𝑢,𝑣𝐿[0,𝑇);𝑊01,2(𝑞+1)(Ω)𝐿2(𝑝+2),𝑢(Ω)𝑡𝐿[0,𝑇);𝐿2(Ω)𝐿𝑚+1𝑣(Ω×(0,𝑇)),𝑡𝐿[0,𝑇);𝐿2(Ω)𝐿𝑟+1(Ω×(0,𝑇)),𝑢𝑡𝑡,𝑣𝑡𝑡𝐿[0,𝑇),𝐿2.(Ω)(2.9)

Define the energy functional 𝐸(𝑡) associated with our system 1𝐸(𝑡)=2𝑢𝑡(𝑡)22+𝑣𝑡(𝑡)22+12Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥Ω𝐹(𝑢,𝑣)𝑑𝑥.(2.10) A simple computation gives 𝑑𝐸(𝑡)𝑑𝑡=𝑢𝑚+1𝑚+1𝑣𝑟+1𝑟+10.(2.11)

Our main result reads as follows.

Theorem 2.2. Assume that (2.3) holds. Assume further that 2(𝑝+2)>max{2𝑞+2,𝑚+1,𝑟+1}. Then any solution of (1.1) with initial data satisfying Ω𝐺||𝑢0||2||+𝐺𝑣0||2𝑑𝑥1/2>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(𝑡) be a solution of the ordinary differential inequality 𝑑Θ(𝑡)𝑑𝑡𝐶Θ1+𝜀(𝑡),𝑡>0,(3.1) where 𝜀>0. If Θ(0)>0, then the solution ceases to exist for 𝑡Θ𝜀(0)𝐶1𝜀1.

Lemma 3.2. Assume that (2.3) holds. Then there exists 𝜂>0 such that for any (𝑢,𝑣)𝑍, one has 𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2𝜂Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥𝑝+2.(3.2)

Proof. By using Minkowski's inequality, we get 𝑢+𝑣22(𝑝+2)2𝑢22(𝑝+2)+𝑣22(𝑝+2).(3.3) Also, Hölder's and Young's inequalities give us 𝑢𝑣𝑝+2𝑢2(𝑝+2)𝑣2(𝑝+2)12𝑢22(𝑝+2)+𝑣22(𝑝+2).(3.4) If 𝑏1>0, then we have Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥𝑐𝑢22+𝑣22.(3.5) If 𝑏1=0, from 𝑏1+𝑏2>0, we have 𝑏2>0. Since 𝑊01,2(𝑞+1)(Ω)𝐻10(Ω), we have 𝑢22+𝑣22𝑐1𝑢22(𝑞+1)+𝑣22(𝑞+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(Ω)𝐿2(𝑝+2)(Ω), we have (3.2).

In order to prove our result and for the sake of simplicity, we take 𝑎=𝑏=1 and introduce the following: 𝐵=𝜂1/(2(𝑝+2)),𝛼1=𝐵(𝑝+2/(𝑝+1)),𝐸1=121𝛼2(𝑝+2)21,𝐸2=112(𝑞+1)𝛼2(𝑝+2)21,(3.7) where 𝜂 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 (𝑢,𝑣)𝑍 be the solution of the system (1.1). Assume further that 𝐸(0)<𝐸1 and Ω𝐺||𝑢0||2||+𝐺𝑣0||2𝑑𝑥1/2>𝛼1.(3.8) Then there exists a constant 𝛼2>𝛼1 such that Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥1/2𝛼2,for𝑡>0,(3.9)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+21/(2(𝑝+2))𝐵𝛼2,for𝑡>0.(3.10)

Proof. We first note that, by (2.10), (3.2), and the definition of 𝐵, we have 1𝐸(𝑡)2Ω𝐺||||𝑢2||||+𝐺𝑣21𝑑𝑥2(𝑝+2)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+212Ω𝐺||||𝑢2||||+𝐺𝑣2𝐵𝑑𝑥2(𝑝+2)2(𝑝+2)Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥𝑝+2=12𝛼2𝐵2(𝑝+2)𝛼2(𝑝+2)2(𝑝+2),(3.11) where 𝛼=(Ω(𝐺(|𝑢|2)+𝐺(|𝑣|2))𝑑𝑥)1/2. It is not hard to verify that 𝑔 is increasing for 0<𝛼<𝛼1, decreasing for 𝛼>𝛼1, 𝑔(𝛼) as 𝛼+, and 𝑔𝛼1=12𝛼21𝐵2(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=(Ω(𝐺(|𝑢0|2)+𝐺(|𝑣0|2))𝑑𝑥)1/2. Then by (3.11) we get 𝑔(𝛼0)𝐸(0)=𝑔(𝛼2), which implies that 𝛼0𝛼2. Now, to establish (3.9), we suppose by contradiction that Ω𝐺||𝑡𝑢0||2||𝑡+𝐺𝑣0||2𝑑𝑥1/2<𝛼2,(3.13) for some 𝑡0>0. By the continuity of Ω(𝐺(|𝑢|2)+𝐺(|𝑣|2))𝑑𝑥, we can choose 𝑡0 such that Ω𝐺||𝑡𝑢0||2||𝑡+𝐺𝑣0||2𝑑𝑥1/2>𝛼1.(3.14) Again, the use of (3.11) leads to 𝐸𝑡0𝑔Ω𝐺||𝑡𝑢0||2||𝑡+𝐺𝑣0||2𝑑𝑥1/2𝛼>𝑔2=𝐸(0).(3.15) This is impossible since 𝐸(𝑡)𝐸(0) for all 𝑡[0,𝑇). Hence (3.9) is established.
To prove (3.10), we make use of (2.10) to get 12Ω𝐺||||𝑢2||||+𝐺𝑣21𝑑𝑥𝐸(0)+2(𝑝+2)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2.(3.16) Consequently, (3.9) yields 12(𝑝+2)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+212Ω𝐺||||𝑢2||||+𝐺𝑣21𝑑𝑥𝐸(0)2𝛼221𝐸(0)2𝛼22𝛼𝑔2=𝐵2(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(𝑡)=𝐸2𝐸(𝑡).(3.18) By using (2.10) and (3.18), we have 0<𝐻(0)𝐻(𝑡)=𝐸212𝑢𝑡(𝑡)22+𝑣𝑡(𝑡)2212Ω𝐺||||𝑢2||||+𝐺𝑣2+1𝑑𝑥2(𝑝+2)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2.(3.19) From (3.9), we have 𝐸212𝑢𝑡(𝑡)22+𝑣𝑡(𝑡)2212Ω𝐺||||𝑢2||||+𝐺𝑣2𝑑𝑥𝐸212𝛼21𝐸112𝛼211=𝛼2(𝑝+2)21<0,𝑡0.(3.20) Hence, by the above inequality and (2.6), we have 10<𝐻(0)𝐻(𝑡)2(𝑝+2)𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2,𝑐(3.21)12(𝑝+2)𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2).(3.22) We then define Θ(𝑡)=𝐻1𝛿(𝑡)+𝜖Ω𝑢𝑢𝑡+𝑣𝑣𝑡𝑑𝑥,(3.23) where 𝜖 small enough is to be chosen later and 0<𝛿min𝑝+1,2(𝑝+2)2(𝑝+2)(𝑚+1),2𝑚(𝑝+2)2(𝑝+2)(𝑟+1)2𝑟(𝑝+2).(3.24) Our goal is to show that Θ(𝑡) satisfies the differential inequality (3.1) which leads to a blow-up in finite time. By taking a derivative of (3.23), we get Θ(𝑡)=(1𝛿)𝐻𝛿(𝑡)𝐻(𝑢𝑡)+𝜖𝑡22+𝑣𝑡22𝜖Ω𝑔||||𝑢2||||𝑢2||||+𝑔𝑣2||||𝑣2𝑑𝑥𝜖Ω||𝑢𝑡||𝑚1𝑢𝑡||𝑣𝑢+𝑡||𝑟1𝑣𝑡𝑣𝑑𝑥+𝜖Ω𝑢𝑓1(𝑢,𝑣)+𝑣𝑓2(𝑢,𝑣)𝑑𝑥=(1𝛿)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖𝑡22+𝑣𝑡22𝑏1𝜖𝑢22+𝑣22𝜖𝑏2𝑢2(𝑞+2)2(𝑞+2)𝜖𝑏2𝑣2(𝑞+2)2(𝑞+2)𝜖Ω||𝑢𝑡||𝑚1𝑢𝑡||𝑣𝑢+𝑡||𝑟1𝑣𝑡𝑣𝑑𝑥+𝜖𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2.(3.25) From the definition of 𝐻(𝑡), it follows that 𝑏2𝑢2(𝑞+2)2(𝑞+2)𝑏2𝑣2(𝑞+2)2(𝑞+2)=2(𝑞+1)𝐻(𝑡)2(𝑞+1)𝐸2𝑢+(𝑞+1)𝑡22+𝑣𝑡22+(𝑞+1)𝑏1𝑢22+𝑣222(𝑞+1)Ω𝐹(𝑢,𝑣)𝑑𝑥,(3.26) which together with (3.25) gives Θ(𝑡)=(1𝛿)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝑏1𝑞𝜖𝑢22+𝑣22𝜖Ω||𝑢𝑡||𝑚1𝑢𝑡||𝑣𝑢+𝑡||𝑟1𝑣𝑡𝑣𝑑𝑥+𝜖1𝑞+1𝑝+2𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2+2(𝑞+1)𝐻(𝑡)2(𝑞+1)𝐸2.(3.27) Then, using (3.10), we obtain Θ(𝑡)(1𝛿)𝐻𝛿𝑢(𝑡)𝐻(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝑏1𝑞𝜖𝑢22+𝑣22+2(𝑞+1)𝐻(𝑡)+𝜖𝑐𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2𝜖Ω||𝑢𝑡||𝑚1𝑢𝑡||𝑣𝑢+𝑡||𝑟1𝑣𝑡𝑣𝑑𝑥,(3.28) where 𝑐=1(𝑞+1)/(𝑝+2)2(𝑞+1)𝐸2(𝐵𝛼2)2(𝑝+2). It is clear that 𝑐>0, since 𝛼2>𝐵(𝑝+2)/(𝑝+1). We now exploit Young's inequality to estimate the last two terms on the right side of (3.28) ||||Ω||𝑢𝑡||𝑚1𝑢𝑡||||𝜂𝑢𝑑𝑥1𝑚+1𝑚+1𝑢𝑚+1𝑚+1+𝑚𝜂1((𝑚+1)/𝑚)𝑢𝑚+1𝑡𝑚+1𝑚+1,||||Ω||𝑣𝑡||𝑟1𝑣𝑡||||𝜂𝑣𝑑𝑥2𝑟+1𝑟+1𝑣𝑟+1𝑟+1+𝑟𝜂2((𝑟+1)/𝑟)𝑣𝑟+1𝑡𝑟+1𝑟+1,(3.29) where 𝜂1,𝜂2 are parameters depending on the time 𝑡 and specified later. Inserting the last two estimates into (3.28), we have Θ(𝑡)(1𝛿)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝑏1𝑞𝜖𝑢22+𝑣22+2(𝑞+1)𝐻(𝑡)+𝜖𝑐𝑢+𝑣2(𝑝+2)2(𝑝+2)+2𝑢𝑣𝑝+2𝑝+2𝜂𝜖1𝑚+1𝑚+1𝑢𝑚+1𝑚+1𝜖𝑚𝜂1((𝑚+1)/𝑚)𝑢𝑚+1𝑡𝑚+1𝑚+1𝜂𝜖2𝑟+1𝑟+1𝑣𝑟+1𝑟+1𝜖𝑟𝜂2((𝑟+1)/𝑟)𝑣𝑟+1𝑡𝑟+1𝑟+1.(3.30) By choosing 𝜂1 and 𝜂2 such that 𝜂1(𝑚+1)/𝑚=𝑀1𝐻𝛿(𝑡),𝜂2(𝑟+1)/𝑟=𝑀2𝐻𝛿(𝑡),(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 Θ(𝑡)((1𝛿)𝑀𝜖)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝑏1𝑞𝜖𝑢22+𝑣22+2(𝑞+1)𝐻(𝑡)+𝜖𝑐2𝑢2(𝑝+2)2(𝑝+2)+2𝑣2(𝑝+2)2(𝑝+2)𝜖𝑀1𝑚𝐻𝛿𝑚(𝑡)𝑢𝑚+1𝑚+1𝜖𝑀2𝑟𝐻𝛿𝑟(𝑡)𝑣𝑟+1𝑟+1,(3.32) where 𝑀=𝑚/(𝑚+1)𝑀1+𝑟/(𝑟+1)𝑀2 and 𝑐2 is a positive constant.
Since 2(𝑝+2)>max{𝑚+1,𝑟+1}, taking into account (2.6) and (3.21), then we have 𝐻𝛿𝑚(𝑡)𝑢𝑚+1𝑚+1𝑐3𝑢2𝛿𝑚(𝑝+2)+(𝑚+1)2(𝑝+2)+𝑣2𝛿𝑚(𝑝+2)2(𝑝+2)𝑢𝑚+1𝑚+1,𝐻𝛿𝑟(𝑡)𝑣𝑟+1𝑟+1𝑐4𝑣2𝛿𝑟(𝑝+2)+(𝑟+1)2(𝑝+2)+𝑢2𝛿𝑟(𝑝+2)2(𝑝+2)𝑣𝑟+1𝑟+1,(3.33) for some positive constants 𝑐3 and 𝑐4. By using (3.24) and the algebraic inequality 𝑧𝜈1𝑧+11+𝑎(𝑧+𝑎),𝑧0,0<𝜈1,𝑎0,(3.34) we have 𝑢2𝛿𝑚(𝑝+2)+(𝑚+1)2(𝑝+2)𝑑𝑢2(𝑝+2)2(𝑝+2)+𝐻(0)𝑑𝑢2(𝑝+2)2(𝑝+2)+𝐻(𝑡),𝑡0,(3.35) where 𝑑=1+1/𝐻(0). Similarly, 𝑣2𝛿𝑟(𝑝+2)+(𝑟+1)2(𝑝+2)𝑑𝑣2(𝑝+2)2(𝑝+2)+𝐻(𝑡),𝑡0.(3.36) Also, since (𝑋+𝑌)𝑠𝐶(𝑋𝑠+𝑌𝑠),𝑋,𝑌0,𝑠>0,(3.37) by using (3.24) and (3.34), we conclude that 𝑣2𝛿𝑚(𝑝+2)2(𝑝+2)𝑢𝑚+1𝑚+1𝐶𝑣2(𝑝+2)2(𝑝+2)+𝑢2(𝑝+2)(𝑚+1)𝐶𝑣2(𝑝+2)2(𝑝+2)+𝑢2(𝑝+2)2(𝑝+2),𝑢2𝛿𝑟(𝑝+2)2(𝑝+2)𝑣𝑟+1𝑟+1𝐶𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)(𝑟+1)𝐶𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2),(3.38) where 𝐶 is a generic positive constant. Taking into account (3.33)–(3.38), estimate (3.32) takes the form Θ(𝑡)((1𝛿)𝑀𝜖)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝜖2(𝑞+1)𝐶1𝑀1𝑚𝐶1𝑀2𝑟𝑐𝐻(𝑡)+𝜖2𝐶2𝑀1𝑚𝐶2𝑀2𝑟𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2),(3.39) where 𝐶1=max{𝑐3𝑑+𝐶,𝑐4𝑑+𝐶}, 𝐶2=max{𝑐3𝑑,𝑐4𝑑}. At this point, and for large values of 𝑀1 and 𝑀2, we can find positive constants 𝜅1 and 𝜅2 such that (3.39) becomes Θ(𝑡)((1𝛿)𝑀𝜖)𝐻𝛿(𝑡)𝐻𝑢(𝑡)+𝜖(𝑞+2)𝑡22+𝑣𝑡22+𝜖𝜅1𝐻(𝑡)+𝜖𝜅2𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2).(3.40) Once 𝑀1 and 𝑀2 are fixed, we pick 𝜖 small enough so that (1𝛿)𝑀𝜖0 and Θ(0)=𝐻1𝛿(0)+𝜖Ω𝑢0𝑢1+𝑣0𝑣1𝑑𝑥>0.(3.41) Since 𝐻(𝑡)0, there exists Λ>0 such that (3.40) becomes Θ𝑢(𝑡)𝜖Λ𝐻(𝑡)+𝑡22+𝑣𝑡22+𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2).(3.42) Then, we have Θ(𝑡)Θ(0),𝑡0.(3.43)
Next, we have by Hölder's and Young's inequalities Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡𝑑𝑥1/(1𝛿)𝐶𝑢𝜏/(1𝛿)2(𝑝+2)+𝑢𝑡2𝑠/(1𝛿)+𝑣𝜏/(1𝛿)2(𝑝+2)+𝑣𝑡2𝑠/(1𝛿),(3.44) for 1/𝜏+1/𝑠=1. We take 𝑠=2(1𝛿), to get 𝜏/(1𝛿)=2/(12𝛿). Here and in the sequel, 𝐶 denotes a positive constant which may change from line to line. By using (3.24) and (3.34), we have 𝑢2/(12𝛿)2(𝑝+2)𝑑𝑢2(𝑝+2)2(𝑝+2)+𝐻(𝑡),𝑣2/(12𝛿)2(𝑝+2)𝑑𝑣2(𝑝+2)2(𝑝+2)+𝐻(𝑡),𝑡0.(3.45) Therefore, (3.44) becomes Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡𝑑𝑥1/(1𝛿)𝐶𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2)+𝑢𝑡22+𝑣𝑡22.(3.46) Note that Θ1/(1𝛿)𝐻(𝑡)=1𝛿(𝑡)+𝜖Ω𝑢𝑢𝑡+𝑣𝑣𝑡𝑑𝑥1/(1𝛿)||||𝐶𝐻(𝑡)+Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡||||𝑑𝑥1/(1𝛿)𝐶𝐻(𝑡)+𝑢2(𝑝+2)2(𝑝+2)+𝑣2(𝑝+2)2(𝑝+2)+𝑢𝑡22+𝑣𝑡22.(3.47) Combining (3.42) with (3.47), we have Θ(𝑡)𝐶Θ1/(1𝛿)(𝑡),𝑡0.(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.


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