Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article
Special Issue

Recent Progress in Differential and Difference Equations

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Research Article | Open Access

Volume 2011 |Article ID 760209 | 14 pages | 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 Diblík
Received26 Nov 2010
Revised05 Jun 2011
Accepted27 Jun 2011
Published11 Aug 2011

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 𝑢𝑡𝑡𝑔||||−div∇𝑢2+||𝑢∇𝑢𝑡||𝑚−1𝑢𝑡=𝑓1𝑣(𝑢,𝑣),(𝑥,𝑡)∈Ω×(0,𝑇),𝑡𝑡𝑔||||−div∇𝑣2+||𝑣∇𝑣𝑡||𝑟−1𝑣𝑡=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),𝑥∈Ω,𝑣(𝑥,0)=𝑣0(𝑥),𝑣𝑡(𝑥,0)=𝑣1(𝑥),𝑥∈Ω,(1.1) where Ω is a bounded domain in ℝ𝑛 with a smooth boundary 𝜕Ω, 𝑚,𝑟≥1, and 𝑓𝑖(⋅,⋅)∶ℝ2→ℝ (𝑖=1,2) are given functions to be specified later. We assume that 𝑔 is a function which satisfies 𝑔∈𝐶1,𝑔(𝑠)>0,𝑔(𝑠)+2ğ‘ ğ‘”î…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 𝑢𝑡𝑡𝑢−Δ𝑢+â„Žğ‘¡î€¸=𝑓(𝑢),𝑥∈Ω,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 ℎ(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 ℎ(𝑢𝑡)=ğ‘Žğ‘¢ğ‘¡ and a polynomial source term of the form 𝑓(𝑢)=𝑏|𝑢|𝑝−2𝑢. 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 ℎ(𝑢𝑡)=ğ‘Ž|𝑢𝑡|𝑚−2𝑢𝑡. In their work, the authors considered problem (1.3) with 𝑓(𝑢)=𝑏|𝑢|𝑝−2𝑢 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 𝑚≥𝑝≥2 and blow up in finite time if 𝑝>𝑚≥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: 𝑢𝑡𝑡𝑢−𝑔𝑥𝑥−𝑢𝑥𝑥𝑡||𝑢+𝛿𝑡||𝑚−1𝑢𝑡=𝜇|𝑢|𝑝−1𝑢,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 [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 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|𝑚−1𝑢𝑡 and 𝜇|𝑢|𝑝−1𝑢 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 𝑢𝑡𝑡𝑔||||−div∇𝑢2𝑢∇𝑢+â„Žğ‘¡î€¸=𝑓(𝑢),𝑥∈Ω,𝑡>0(1.5) with initial and Dirichlet boundary condition, where 𝑔 satisfies condition (1.2) and 𝑔(𝑠)≥𝑏1+𝑏2ğ‘ ğ‘ž,ğ‘žâ‰¥0.(1.6) The damping term has the form â„Žî€·ğ‘¢ğ‘¡î€¸=𝑑1𝑢𝑡+𝑑2||𝑢𝑡||𝑟−1𝑢𝑡,𝑟>1.(1.7) The source term is 𝑓(𝑢)=ğ‘Ž1𝑢+ğ‘Ž2|𝑢|𝑝−1𝑢(1.8) with 𝑝≥1 for 𝑛=1,2 and 1≤𝑛≤2𝑛/(𝑛−2) for 𝑛≥3, ğ‘Ž1,ğ‘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(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 ||⋅||ğ‘ž the ğ¿ğ‘ž(Ω) norm for 1â‰¤ğ‘žâ‰¤âˆž 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)=ğ‘Ž|𝑢+𝑣|2(𝑝+1)(𝑢+𝑣)+𝑏|𝑢|𝑝𝑢|𝑣|(𝑝+2),𝑓2î€ºğ‘Ž(𝑢,𝑣)=|𝑢+𝑣|2(𝑝+1)(𝑢+𝑣)+𝑏|𝑢|(𝑝+2)|𝑣|𝑝𝑣,(2.2) where ğ‘Ž,𝑏>0 are constants and 𝑝 satisfies âŽ§âŽªâŽ¨âŽªâŽ©ğ‘>−1,if𝑛=1,2,−1<𝑝≤4−𝑛𝑛−2,if𝑛≥3.(2.3) One can easily verify that 𝑢𝑓1(𝑢,𝑣)+𝑣𝑓2(𝑢,𝑣)=2(𝑝+2)𝐹(𝑢,𝑣),∀(𝑢,𝑣)∈ℝ2,(2.4) where 1𝐹(𝑢,𝑣)=2(𝑝+2)ğ‘Ž|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 𝑐02(𝑝+2)|𝑢|2(𝑝+2)+|𝑣|2(𝑝+2)𝑐≤𝐹(𝑢,𝑣)≤12(𝑝+2)|𝑢|2(𝑝+2)+|𝑣|2(𝑝+2).(2.6)

Throughout this paper, we define 𝑔 by 𝑔(𝑠)=𝑏1+𝑏2ğ‘ ğ‘ž,ğ‘žâ‰¥0,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,𝑠≥0.(2.8)

In order to state and prove our result, we introduce the following function space: 𝑍=(𝑢,𝑣)∣𝑢,ğ‘£âˆˆğ¿âˆžî‚€[0,𝑇);𝑊01,2(ğ‘ž+1)(Ω)∩𝐿2(𝑝+2),𝑢(Ω)ğ‘¡âˆˆğ¿âˆžî€·[0,𝑇);𝐿2(Ω)∩𝐿𝑚+1𝑣(Ω×(0,𝑇)),ğ‘¡âˆˆğ¿âˆžî€·[0,𝑇);𝐿2(Ω)∩𝐿𝑟+1(Ω×(0,𝑇)),𝑢𝑡𝑡,ğ‘£ğ‘¡ğ‘¡âˆˆğ¿âˆžî€·[0,𝑇),𝐿2.(Ω)(2.9)

Define the energy functional 𝐸(𝑡) associated with our system 1𝐸(𝑡)=2‖‖𝑢𝑡(‖‖𝑡)22+‖‖𝑣𝑡(‖‖𝑡)22+12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥−Ω𝐹(𝑢,𝑣)𝑑𝑥.(2.10) A simple computation gives 𝑑𝐸(𝑡)𝑑𝑡=−‖𝑢‖𝑚+1𝑚+1−‖𝑣‖𝑟+1𝑟+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{2ğ‘ž+2,𝑚+1,𝑟+1}. Then any solution of (1.1) with initial data satisfying Ω𝐺||∇𝑢0||2||+𝐺∇𝑣0||2𝑑𝑥1/2>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)𝑑𝑡≥𝐶Θ1+𝜀(𝑡),𝑡>0,(3.1) where 𝜀>0. If Θ(0)>0, then the solution ceases to exist for 𝑡≥Θ−𝜀(0)𝐶−1𝜀−1.

Lemma 3.2. Assume that (2.3) holds. Then there exists 𝜂>0 such that for any (𝑢,𝑣)∈𝑍, one has ‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2≤𝜂Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥𝑝+2.(3.2)

Proof. By using Minkowski's inequality, we get ‖𝑢+𝑣‖22(𝑝+2)‖≤2𝑢‖22(𝑝+2)+‖𝑣‖22(𝑝+2).(3.3) Also, Hölder's and Young's inequalities give us ‖𝑢𝑣‖𝑝+2≤‖𝑢‖2(𝑝+2)‖𝑣‖2(𝑝+2)≤12‖𝑢‖22(𝑝+2)+‖𝑣‖22(𝑝+2).(3.4) If 𝑏1>0, then we have Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥≥𝑐‖∇𝑢‖22+‖∇𝑣‖22.(3.5) If 𝑏1=0, from 𝑏1+𝑏2>0, we have 𝑏2>0. Since 𝑊01,2(ğ‘ž+1)(Ω)↪𝐻10(Ω), we have ‖∇𝑢‖22+‖∇𝑣‖22≤𝑐1‖∇𝑢‖22(ğ‘ž+1)+‖∇𝑣‖22(ğ‘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(Ω)↪𝐿2(𝑝+2)(Ω), we have (3.2).

In order to prove our result and for the sake of simplicity, we take ğ‘Ž=𝑏=1 and introduce the following: 𝐵=𝜂1/(2(𝑝+2)),𝛼1=𝐵−(𝑝+2/(𝑝+1)),𝐸1=12−1𝛼2(𝑝+2)21,𝐸2=1−12(ğ‘ž+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 Ω𝐺||∇𝑢0||2||+𝐺∇𝑣0||2𝑑𝑥1/2>𝛼1.(3.8) Then there exists a constant 𝛼2>𝛼1 such that Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥1/2≥𝛼2,for𝑡>0,(3.9)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+21/(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)≥2Ω𝐺||||∇𝑢2||||+𝐺∇𝑣21𝑑𝑥−2(𝑝+2)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2≥12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝐵𝑑𝑥−2(𝑝+2)2(𝑝+2)Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥𝑝+2=12𝛼2−𝐵2(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 𝑔𝛼1=12𝛼21−𝐵2(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 𝛼0≥𝛼2. Now, to establish (3.9), we suppose by contradiction that Ω𝐺||𝑡∇𝑢0||2||𝑡+𝐺∇𝑣0||2𝑑𝑥1/2<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 Ω𝐺||𝑡∇𝑢0||2||𝑡+𝐺∇𝑣0||2𝑑𝑥1/2>𝛼1.(3.14) Again, the use of (3.11) leads to 𝐸𝑡0≥𝑔Ω𝐺||𝑡∇𝑢0||2||𝑡+𝐺∇𝑣0||2𝑑𝑥1/2𝛼>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 12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣21𝑑𝑥≤𝐸(0)+2(𝑝+2)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2.(3.16) Consequently, (3.9) yields 12(𝑝+2)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2≥12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2≥1𝑑𝑥−𝐸(0)2𝛼221−𝐸(0)≥2𝛼22𝛼−𝑔2=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)=𝐸2−𝐸(𝑡).(3.18) By using (2.10) and (3.18), we have 0<𝐻(0)≤𝐻(𝑡)=𝐸2−12‖‖𝑢𝑡(‖‖𝑡)22+‖‖𝑣𝑡(‖‖𝑡)22−12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2+1𝑑𝑥2(𝑝+2)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2.(3.19) From (3.9), we have 𝐸2−12‖‖𝑢𝑡(‖‖𝑡)22+‖‖𝑣𝑡(‖‖𝑡)22−12Ω𝐺||||∇𝑢2||||+𝐺∇𝑣2𝑑𝑥≤𝐸2−12𝛼21≤𝐸1−12𝛼211=−𝛼2(𝑝+2)21<0,∀𝑡≥0.(3.20) Hence, by the above inequality and (2.6), we have 10<𝐻(0)≤𝐻(𝑡)≤2(𝑝+2)‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2,≤𝑐(3.21)12(𝑝+2)‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.22) We then define Θ(𝑡)=𝐻1−𝛿(𝑡)+𝜖Ω𝑢𝑢𝑡+𝑣𝑣𝑡𝑑𝑥,(3.23) where 𝜖 small enough is to be chosen later and 0<𝛿≤min𝑝+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)=(1−𝛿)𝐻−𝛿(𝑡)ğ»î…ž(‖‖𝑢𝑡)+𝜖𝑡‖‖22+‖‖𝑣𝑡‖‖22−𝜖Ω𝑔||||∇𝑢2||||∇𝑢2||||+𝑔∇𝑣2||||∇𝑣2𝑑𝑥−𝜖Ω||𝑢𝑡||𝑚−1𝑢𝑡||𝑣𝑢+𝑡||𝑟−1𝑣𝑡𝑣𝑑𝑥+𝜖Ω𝑢𝑓1(𝑢,𝑣)+𝑣𝑓2(𝑢,𝑣)𝑑𝑥=(1−𝛿)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖𝑡‖‖22+‖‖𝑣𝑡‖‖22−𝑏1𝜖‖∇𝑢‖22+‖∇𝑣‖22−𝜖𝑏2‖∇𝑢‖2(ğ‘ž+2)2(ğ‘ž+2)−𝜖𝑏2‖∇𝑣‖2(ğ‘ž+2)2(ğ‘ž+2)−𝜖Ω||𝑢𝑡||𝑚−1𝑢𝑡||𝑣𝑢+𝑡||𝑟−1𝑣𝑡𝑣𝑑𝑥+𝜖‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2.(3.25) From the definition of 𝐻(𝑡), it follows that −𝑏2‖∇𝑢‖2(ğ‘ž+2)2(ğ‘ž+2)−𝑏2‖∇𝑣‖2(ğ‘ž+2)2(ğ‘ž+2)=2(ğ‘ž+1)𝐻(𝑡)−2(ğ‘ž+1)𝐸2‖‖𝑢+(ğ‘ž+1)𝑡‖‖22+‖‖𝑣𝑡‖‖22+(ğ‘ž+1)𝑏1‖∇𝑢‖22+‖∇𝑣‖22−2(ğ‘ž+1)Ω𝐹(𝑢,𝑣)𝑑𝑥,(3.26) which together with (3.25) gives Θ′(𝑡)=(1−𝛿)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝑏1î€·â€–ğ‘žğœ–âˆ‡ğ‘¢â€–22+‖∇𝑣‖22−𝜖Ω||𝑢𝑡||𝑚−1𝑢𝑡||𝑣𝑢+𝑡||𝑟−1𝑣𝑡𝑣𝑑𝑥+𝜖1âˆ’ğ‘ž+1‖𝑝+2𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2+2(ğ‘ž+1)𝐻(𝑡)−2(ğ‘ž+1)𝐸2.(3.27) Then, using (3.10), we obtain Θ′(𝑡)≥(1−𝛿)𝐻−𝛿‖‖𝑢(𝑡)𝐻′(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝑏1î€·â€–ğ‘žğœ–âˆ‡ğ‘¢â€–22+‖∇𝑣‖22+2(ğ‘ž+1)𝐻(𝑡)+𝜖𝑐‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2−𝜖Ω||𝑢𝑡||𝑚−1𝑢𝑡||𝑣𝑢+𝑡||𝑟−1𝑣𝑡𝑣𝑑𝑥,(3.28) where 𝑐=1−(ğ‘ž+1)/(𝑝+2)−2(ğ‘ž+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||𝑚−1𝑢𝑡||||≤𝜂𝑢𝑑𝑥1𝑚+1𝑚+1‖𝑢‖𝑚+1𝑚+1+𝑚𝜂1−((𝑚+1)/𝑚)‖‖𝑢𝑚+1𝑡‖‖𝑚+1𝑚+1,||||Ω||𝑣𝑡||𝑟−1𝑣𝑡||||≤𝜂𝑣𝑑𝑥2𝑟+1𝑟+1‖𝑣‖𝑟+1𝑟+1+𝑟𝜂2−((𝑟+1)/𝑟)‖‖𝑣𝑟+1𝑡‖‖𝑟+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)≥(1−𝛿)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝑏1î€·â€–ğ‘žğœ–âˆ‡ğ‘¢â€–22+‖∇𝑣‖22+2(ğ‘ž+1)𝐻(𝑡)+𝜖𝑐‖𝑢+𝑣‖2(𝑝+2)2(𝑝+2)+2‖𝑢𝑣‖𝑝+2𝑝+2𝜂−𝜖1𝑚+1𝑚+1‖𝑢‖𝑚+1𝑚+1−𝜖𝑚𝜂1−((𝑚+1)/𝑚)‖‖𝑢𝑚+1𝑡‖‖𝑚+1𝑚+1𝜂−𝜖2𝑟+1‖𝑟+1𝑣‖𝑟+1𝑟+1−𝜖𝑟𝜂2−((𝑟+1)/𝑟)‖‖𝑣𝑟+1𝑡‖‖𝑟+1𝑟+1.(3.30) By choosing 𝜂1 and 𝜂2 such that 𝜂1−(𝑚+1)/𝑚=𝑀1𝐻−𝛿(𝑡),𝜂2−(𝑟+1)/𝑟=𝑀2𝐻−𝛿(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)≥((1−𝛿)−𝑀𝜖)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝑏1î€·â€–ğ‘žğœ–âˆ‡ğ‘¢â€–22+‖∇𝑣‖22+2(ğ‘ž+1)𝐻(𝑡)+𝜖𝑐2‖𝑢‖2(𝑝+2)2(𝑝+2)+2‖𝑣‖2(𝑝+2)2(𝑝+2)−𝜖𝑀1−𝑚𝐻𝛿𝑚(𝑡)‖𝑢‖𝑚+1𝑚+1−𝜖𝑀2−𝑟𝐻𝛿𝑟(𝑡)‖𝑣‖𝑟+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)𝑢‖𝑚+1𝑚+1≤𝑐3‖𝑢‖2𝛿𝑚(𝑝+2)+(𝑚+1)2(𝑝+2)+‖𝑣‖2𝛿𝑚(𝑝+2)2(𝑝+2)‖𝑢‖𝑚+1𝑚+1,𝐻𝛿𝑟(𝑡)‖𝑣‖𝑟+1𝑟+1≤𝑐4‖𝑣‖2𝛿𝑟(𝑝+2)+(𝑟+1)2(𝑝+2)+‖𝑢‖2𝛿𝑟(𝑝+2)2(𝑝+2)‖𝑣‖𝑟+1𝑟+1,(3.33) for some positive constants 𝑐3 and 𝑐4. By using (3.24) and the algebraic inequality 𝑧𝜈1≤𝑧+1≤1+ğ‘Žî‚(𝑧+ğ‘Ž),∀𝑧≥0,0<𝜈≤1,ğ‘Žâ‰¥0,(3.34) we have ‖𝑢‖2𝛿𝑚(𝑝+2)+(𝑚+1)2(𝑝+2)‖≤𝑑𝑢‖2(𝑝+2)2(𝑝+2)‖+𝐻(0)≤𝑑𝑢‖2(𝑝+2)2(𝑝+2)+𝐻(𝑡),∀𝑡≥0,(3.35) where 𝑑=1+1/𝐻(0). Similarly, ‖𝑣‖2𝛿𝑟(𝑝+2)+(𝑟+1)2(𝑝+2)‖≤𝑑𝑣‖2(𝑝+2)2(𝑝+2)+𝐻(𝑡),∀𝑡≥0.(3.36) Also, since (𝑋+𝑌)𝑠≤𝐶(𝑋𝑠+𝑌𝑠),𝑋,𝑌≥0,𝑠>0,(3.37) by using (3.24) and (3.34), we conclude that ‖𝑣‖2𝛿𝑚(𝑝+2)2(𝑝+2)‖𝑢‖𝑚+1𝑚+1‖≤𝐶𝑣‖2(𝑝+2)2(𝑝+2)+‖𝑢‖2(𝑝+2)(𝑚+1)‖≤𝐶𝑣‖2(𝑝+2)2(𝑝+2)+‖𝑢‖2(𝑝+2)2(𝑝+2),‖𝑢‖2𝛿𝑟(𝑝+2)2(𝑝+2)‖𝑣‖𝑟+1𝑟+1≤𝐶‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)(𝑟+1)≤𝐶‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(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)≥((1−𝛿)−𝑀𝜖)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝜖2(ğ‘ž+1)−𝐶1𝑀1−𝑚−𝐶1𝑀2−𝑟𝑐𝐻(𝑡)+𝜖2−𝐶2𝑀1−𝑚−𝐶2𝑀2−𝑟‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(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)≥((1−𝛿)−𝑀𝜖)𝐻−𝛿(𝑡)ğ»î…žî‚€â€–â€–ğ‘¢(𝑡)+𝜖(ğ‘ž+2)𝑡‖‖22+‖‖𝑣𝑡‖‖22+𝜖𝜅1𝐻(𝑡)+𝜖𝜅2‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.40) Once 𝑀1 and 𝑀2 are fixed, we pick 𝜖 small enough so that (1−𝛿)−𝑀𝜖≥0 and Θ(0)=𝐻1−𝛿(0)+𝜖Ω𝑢0𝑢1+𝑣0𝑣1𝑑𝑥>0.(3.41) Since ğ»î…ž(𝑡)≥0, there exists Λ>0 such that (3.40) becomes Î˜î…žî‚€â€–â€–ğ‘¢(𝑡)≥𝜖Λ𝐻(𝑡)+𝑡‖‖22+‖‖𝑣𝑡‖‖22+‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2).(3.42) Then, we have Θ(𝑡)≥Θ(0),∀𝑡≥0.(3.43)
Next, we have by Hölder's and Young's inequalities Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡𝑑𝑥1/(1−𝛿)‖≤𝐶𝑢‖𝜏/(1−𝛿)2(𝑝+2)+‖‖𝑢𝑡‖‖2𝑠/(1−𝛿)+‖𝑣‖𝜏/(1−𝛿)2(𝑝+2)+‖‖𝑣𝑡‖‖2𝑠/(1−𝛿),(3.44) for 1/𝜏+1/𝑠=1. We take 𝑠=2(1−𝛿), to get 𝜏/(1−𝛿)=2/(1−2𝛿). 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 ‖𝑢‖2/(1−2𝛿)2(𝑝+2)‖≤𝑑𝑢‖2(𝑝+2)2(𝑝+2)+𝐻(𝑡),‖𝑣‖2/(1−2𝛿)2(𝑝+2)‖≤𝑑𝑣‖2(𝑝+2)2(𝑝+2)+𝐻(𝑡),∀𝑡≥0.(3.45) Therefore, (3.44) becomes Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡𝑑𝑥1/(1−𝛿)‖≤𝐶𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2)+‖‖𝑢𝑡‖‖22+‖‖𝑣𝑡‖‖22.(3.46) Note that Θ1/(1−𝛿)𝐻(𝑡)=1−𝛿(𝑡)+𝜖Ω𝑢𝑢𝑡+𝑣𝑣𝑡𝑑𝑥1/(1−𝛿)||||≤𝐶𝐻(𝑡)+Ω𝑢𝑢𝑡𝑑𝑥+Ω𝑣𝑣𝑡||||𝑑𝑥1/(1−𝛿)≤𝐶𝐻(𝑡)+‖𝑢‖2(𝑝+2)2(𝑝+2)+‖𝑣‖2(𝑝+2)2(𝑝+2)+‖‖𝑢𝑡‖‖22+‖‖𝑣𝑡‖‖22.(3.47) Combining (3.42) with (3.47), we have Θ(𝑡)≥𝐶Θ1/(1−𝛿)(𝑡),∀𝑡≥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.

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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.

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