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

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 𝑢𝑣_𝑡𝑡−div𝑔||∇𝑢||²∇𝑢+||𝑢^𝑡||^𝑚−1𝑢_𝑡=𝑓₁(𝑢,𝑣),(𝑥,𝑡)∈Ω×(0,𝑇),_𝑡𝑡−div𝑔𝑢(𝑥||∇𝑢,0(𝑣𝑥)||²,=𝑡)∇𝑢=₀𝑣𝑣(+𝑥,||𝑣𝑡)_𝑡||=^𝑟−10𝑣,_𝑡=𝑥𝑓∈₂(𝜕𝑢Ω,𝑣×),(0(,𝑥𝑇,)𝑡),∈Ω×(0,𝑇),𝑣(𝑥,0)=𝑣₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥),𝑥∈Ω,(𝑥),𝑣_𝑡(𝑥,0)=𝑣₁(𝑥),𝑥∈Ω,(1.1) where Ω is a bounded domain in ℝ^𝑛 with a smooth boundary 𝜕Ω, 𝑚,𝑟≥1, and 𝑓_𝑖(⋅,⋅)∶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)+2𝑠𝑔^(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 𝑢_𝑡𝑡−Δ𝑢+ℎ𝑢_𝑡=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(𝑢_𝑡) 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(𝑢_𝑡)=𝑎𝑢_𝑡 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(𝑢_𝑡)=𝑎|𝑢_𝑡|^𝑚−2𝑢_𝑡. 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: 𝑢_𝑡𝑡−𝑔𝑢_𝑥_𝑥−𝑢_𝑥𝑥𝑡+𝛿||𝑢_𝑡||^𝑚−1𝑢_𝑡=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|𝑢_𝑡|^𝑚−1𝑢_𝑡 and 𝜇|𝑢|^𝑝−1𝑢 by 𝑔(𝑢_𝑡) 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 𝑢_𝑡𝑡−div𝑔||∇𝑢||²∇𝑢+ℎ𝑢^𝑡=𝑓(𝑢),𝑥∈Ω,𝑡>0(1.5) with initial and Dirichlet boundary condition, where 𝑔 satisfies condition (1.2) and 𝑔(𝑠)≥𝑏₁+𝑏₂𝑠^𝑞,𝑞≥0.(1.6) The damping term has the form ℎ𝑢_𝑡=𝑑₁𝑢_𝑡+𝑑₂||𝑢_𝑡||^𝑟−1𝑢_𝑡,𝑟>1.(1.7) The source term is 𝑓(𝑢)=𝑎₁𝑢+𝑎₂|𝑢|^𝑝−1𝑢(1.8) with 𝑝≥1 for 𝑛=1,2 and 1≤𝑛≤2𝑛/(𝑛−2) for 𝑛≥3, 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 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 ||⋅||_𝑞 the 𝐿^𝑞(Ω) norm for 1≤𝑞≤∞ 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 𝑓𝑓₁₂(𝑢,𝑣)=𝑎|𝑢+𝑣|^2(𝑝+1)(𝑢+𝑣)+𝑏|𝑢|^𝑝𝑢|𝑣|^(𝑝+2),(𝑢,𝑣)=𝑎|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><−1𝑝,≤4−𝑛if𝑛=1,2,𝑛−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 𝐹(𝑢,𝑣)=12(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)≤𝐹(𝑢,𝑣)≤𝑐¹2(𝑝+2)|𝑢|^2(𝑝+2)+|𝑣|^2(𝑝+2).(2.6)

Throughout this paper, we define 𝑔 by 𝑔(𝑠)=𝑏₁+𝑏₂𝑠^𝑞,𝑞≥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(𝑢,𝑣)∣𝑢,𝑣∈𝐿^∞[0,𝑇);𝑊^01,2(𝑞+1)(Ω)∩𝐿^2(𝑝+2)(Ω)_𝑡,𝑣∈𝐿^∞[0,𝑇);𝐿²(Ω)∩𝐿^𝑚+1(Ω×(0,𝑇)),_𝑡∈𝐿^∞[0,𝑇);𝐿²(Ω)∩𝐿^𝑟+1(Ω×(0,𝑇)),𝑢_𝑡𝑡,𝑣_𝑡𝑡∈𝐿^∞[0,𝑇),𝐿²(Ω).(2.9)

Define the energy functional 𝐸(𝑡) associated with our system 𝐸(𝑡)=12‖‖𝑢^𝑡(𝑡)‖‖²²+‖‖𝑣_𝑡(𝑡)‖‖²²+12_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²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{2𝑞+2,𝑚+1,𝑟+1}. Then any solution of (1.1) with initial data satisfying _Ω𝐺||∇𝑢⁰||²+𝐺||∇𝑣⁰||²𝑑𝑥^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(𝑡) be a solution of the ordinary differential inequality 𝑑Θ(𝑡)𝑑𝑡≥𝐶Θ^1+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≤𝜂_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥^𝑝+2.(3.2)

Proof. By using Minkowski's inequality, we get ‖𝑢+𝑣‖^22(𝑝+2)≤2‖𝑢‖^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||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥≥𝑐‖∇𝑢‖²²+‖∇𝑣‖²².(3.5) If 𝑏₁=0, from 𝑏₁+𝑏₂>0, we have 𝑏₂>0. Since 𝑊^01,2(𝑞+1)(Ω)↪𝐻¹⁰(Ω), we have ‖∇𝑢‖²²+‖∇𝑣‖²²≤𝑐₁‖∇𝑢‖^22(𝑞+1)+‖∇𝑣‖^22(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)↪𝐿^2(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=𝜂^{1/(2(𝑝+2))},𝛼₁=𝐵^{−(𝑝+2/(𝑝+1))},𝐸₁=12−2(𝑝1+2)𝛼²¹𝐸₂,=12(𝑞+1)−2(𝑝1+2)𝛼²¹,(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𝑑𝑥^1/2>𝛼₁.(3.8) Then there exists a constant 𝛼₂>𝛼₁ such that _Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥^1/2‖𝑢+𝑣‖_{2(𝑝+2)2(𝑝+2)}≥𝛼₂,for𝑡>0,(3.9)+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 𝐸(𝑡)≥12_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥−2(𝑝1+2)‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}≥1+2‖𝑢𝑣‖^𝑝+2𝑝+22_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥−2𝐵(𝑝+2^2(𝑝+2))_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥^𝑝+2=12𝛼²−𝐵^2(𝑝+2)2(𝑝+2)𝛼^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−𝐵^2(𝑝+2)2(𝑝+2)𝛼^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||∇𝑢𝑡⁰||²+𝐺||∇𝑣𝑡⁰||²𝑑𝑥^1/2<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||∇𝑢𝑡⁰||²+𝐺||∇𝑣𝑡⁰||²𝑑𝑥^1/2>𝛼₁.(3.14) Again, the use of (3.11) leads to 𝐸𝑡₀≥𝑔_Ω𝐺||∇𝑢𝑡⁰||²+𝐺||∇𝑣𝑡⁰||²𝑑𝑥^1/2>𝑔𝛼²=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||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥≤𝐸(0)+2(𝑝1+2)‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑢𝑣‖^𝑝+2𝑝+2.(3.16) Consequently, (3.9) yields 12(𝑝+2)‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑢𝑣‖^𝑝+2𝑝+2≥1≥12_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥−𝐸(0)2𝛼²²−𝐸(0)≥12𝛼²²−𝑔𝛼₂=𝐵^2(𝑝+2)2(𝑝+2)𝛼^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‖‖𝑢^𝑡(𝑡)‖‖²²+‖‖𝑣_𝑡(𝑡)‖‖²²−1+12_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥2(𝑝+2)‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑢𝑣‖^𝑝+2𝑝+2.(3.19) From (3.9), we have 𝐸₂−12‖‖𝑢^𝑡(𝑡)‖‖²²+‖‖𝑣_𝑡(𝑡)‖‖²²−1≤𝐸₂2_Ω𝐺||∇𝑢||²+𝐺||∇𝑣||²𝑑𝑥−12𝛼²¹≤𝐸₁−12𝛼²¹=−12(𝑝+2)𝛼²¹<0,∀𝑡≥0.(3.20) Hence, by the above inequality and (2.6), we have 0<𝐻(0)≤𝐻(𝑡)≤12(𝑝+2)‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}≤𝑐+2‖𝑢𝑣‖^𝑝+2𝑝+2,(3.21)₁2(𝑝+2)‖𝑢‖^{2(𝑝+2)2(𝑝+2)}+‖𝑣‖^{2(𝑝+2)2(𝑝+2)}.(3.22) We then define Θ(𝑡)=𝐻^1−𝛿(𝑡)+𝜖_Ω𝑢𝑢_𝑡+𝑣𝑣_𝑡𝑑𝑥,(3.23) where 𝜖 small enough is to be chosen later and 0<𝛿≤min𝑝+12(𝑝+2),2(𝑝+2𝑚2()𝑝−+(𝑚2)+1),2(𝑝+2𝑟2()𝑝−+(2𝑟)+1).(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)=−(1𝜖−_Ω𝛿)𝐻^−𝛿(𝑡)𝐻^(𝑡)+𝜖‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²−𝜖_Ω𝑔||∇𝑢||²||∇𝑢||²+𝑔||∇𝑣||²||∇𝑣||²𝑑𝑥||𝑢^𝑡=(1−𝛿)𝐻||^𝑚−1_−𝛿𝑢_𝑡𝑢+||𝑣_𝑡||^𝑟−1𝑣_𝑡𝑣𝑑𝑥+𝜖_Ω𝑢𝑓₁(𝑢,𝑣)+𝑣𝑓₂(𝑢,𝑣)𝑑𝑥(𝑡)𝐻^(𝑡)+𝜖‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²−𝑏¹𝜖‖∇𝑢‖²²+‖∇𝑣‖²²−𝜖𝑏₂‖∇𝑢‖^{2(𝑞+2)2(𝑞+2)}−𝜖𝑏₂‖∇𝑣‖^{2(𝑞+2)2(𝑞+2)}−𝜖_Ω||𝑢^𝑡||^𝑚−1𝑢_𝑡𝑢+||𝑣_𝑡||^𝑟−1𝑣_𝑡𝑣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(𝑞+2)2(𝑞+2)}−𝑏₂‖∇𝑣‖^{2(𝑞+2)2(𝑞+2)}=+2((𝑞𝑞++11))𝐻𝑏₁(𝑡)−2(𝑞+1)𝐸₂+(𝑞+1)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²‖∇𝑢‖²²+‖∇𝑣‖²²−2(𝑞+1)_Ω𝐹(𝑢,𝑣)𝑑𝑥,(3.26) which together with (3.25) gives Θ′(𝑡)=−(1𝜖−_Ω𝛿)𝐻^−𝛿(𝑡)𝐻^(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²+𝑏¹𝑞𝜖‖∇𝑢‖²²+‖∇𝑣‖²²||𝑢^𝑡||^𝑚−1𝑢_𝑡𝑢+||𝑣_𝑡||^𝑟−1𝑣_𝑡𝑣𝑑𝑥+𝜖1−𝑞+1𝑝+2‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2(𝑞+1)𝐻(𝑡)−2(𝑞+1)𝐸₂+2‖𝑢𝑣‖^𝑝+2𝑝+2.(3.27) Then, using (3.10), we obtain Θ′(𝑡)≥+(1𝜖−𝛿)𝐻^−𝛿(𝑡)𝐻′(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²+𝑏¹𝑞𝜖‖∇𝑢‖²²+‖∇𝑣‖²²+2(𝑞+1)𝐻(𝑡)𝑐‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑢𝑣‖^𝑝+2𝑝+2−𝜖_Ω||𝑢^𝑡||^𝑚−1𝑢_𝑡𝑢+||𝑣_𝑡||^𝑟−1𝑣_𝑡𝑣𝑑𝑥,(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||𝑢_𝑡||^𝑚−1𝑢_𝑡𝑢𝑑𝑥||||≤𝜂^1𝑚+1𝑚+1‖𝑢‖^𝑚+1𝑚+1+𝑚𝜂^{1−((𝑚+1)/𝑚)}𝑚+1‖‖𝑢^𝑡||||_Ω‖‖^𝑚+1𝑚+1,||𝑣_𝑡||^𝑟−1𝑣_𝑡𝑣𝑑𝑥||||≤𝜂^2𝑟+1𝑟+1‖𝑣‖^𝑟+1𝑟+1+𝑟𝜂^{2−((𝑟+1)/𝑟)}𝑟+1‖‖𝑣^𝑡‖‖^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)≥+(1𝜖−𝛿)𝐻^−𝛿(𝑡)𝐻^(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²+𝑏¹𝑞𝜖‖∇𝑢‖²²+‖∇𝑣‖²²+2(𝑞+1)𝐻(𝑡)𝑐‖𝑢+𝑣‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑢𝑣‖^𝑝+2𝑝+2−𝜖𝜂^1𝑚+1𝑚+1‖𝑢‖^𝑚+1𝑚+1−𝜖𝑚𝜂^{1−((𝑚+1)/𝑚)}𝑚+1‖‖𝑢^𝑡‖‖^𝑚+1𝑚+1−𝜖𝜂^2𝑟+1𝑟+1‖𝑣‖^𝑟+1𝑟+1−𝜖𝑟𝜂^{2−((𝑟+1)/𝑟)}𝑟+1‖‖𝑣^𝑡‖‖^𝑟+1𝑟+1.(3.30) By choosing 𝜂₁ and 𝜂₂ such that 𝜂^{1−(𝑚+1)/𝑚}=𝑀₁𝐻^−𝛿(𝑡),𝜂^{2−(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)≥+((21(−𝑞+𝛿)1−)𝐻𝑀(𝜖𝑡))𝐻+𝜖^−𝛿𝑐₂(𝑡)𝐻^(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²+𝑏¹𝑞𝜖‖∇𝑢‖²²+‖∇𝑣‖²²‖𝑢‖^{2(𝑝+2)2(𝑝+2)}+2‖𝑣‖^{2(𝑝+2)2(𝑝+2)}−𝜖𝑀^1−𝑚𝐻^𝛿𝑚(𝑡)‖𝑢‖^𝑚+1𝑚+1−𝜖𝑀^2−𝑟𝐻^𝛿𝑟(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 𝐻𝐻^𝛿𝑚_𝛿𝑟(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,(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 𝑧^𝜈≤𝑧+1≤1+1𝑎(𝑧+𝑎),∀𝑧≥0,0<𝜈≤1,𝑎≥0,(3.34) we have ‖𝑢‖^{2𝛿𝑚(𝑝+2)+(𝑚+1)2(𝑝+2)}≤𝑑‖𝑢‖^{2(𝑝+2)2(𝑝+2)}+𝐻(0)≤𝑑‖𝑢‖^{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)}≤𝑑‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+𝐻(𝑡),∀𝑡≥0.(3.36) Also, since (𝑋+𝑌)^𝑠≤𝐶(𝑋^𝑠+𝑌^𝑠),𝑋,𝑌≥0,𝑠>0,(3.37) by using (3.24) and (3.34), we conclude that ‖‖𝑣𝑢‖‖^{2𝛿𝑚(𝑝+2)2(𝑝+2)}_{2𝛿𝑟(𝑝+2)2(𝑝+2)}‖𝑢‖^𝑚+1𝑚+1≤𝐶‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+‖𝑢‖^{2(𝑝+2)(𝑚+1)}≤𝐶‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+‖𝑢‖^{2(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)≥+((𝜖1−2(𝛿𝑞)+−1𝑀)𝜖−)𝐶𝐻^−𝛿₁(𝑡)𝐻^(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²+𝜖𝑐₂𝑀^1−𝑚−𝐶₁𝑀^2−𝑟𝐻(𝑡)−𝐶₂𝑀^1−𝑚−𝐶₂𝑀^2−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)≥+((𝜖1𝜅−₁𝛿)−𝑀𝜖)𝐻^−𝛿(𝑡)𝐻^(𝑡)+𝜖(𝑞+2)‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²²𝐻(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)=𝐻^1−𝛿(0)+𝜖_Ω𝑢₀𝑢₁+𝑣₀𝑣₁𝑑𝑥>0.(3.41) Since 𝐻^(𝑡)≥0, there exists Λ>0 such that (3.40) becomes Θ^(𝑡)≥𝜖Λ𝐻(𝑡)+‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖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 _Ω𝑢𝑢_𝑡𝑑𝑥+_Ω𝑣𝑣_𝑡𝑑𝑥^1/(1−𝛿)≤𝐶‖𝑢‖^{𝜏/(1−𝛿)2(𝑝+2)}+‖‖𝑢_𝑡‖‖^{2𝑠/(1−𝛿)}+‖𝑣‖^{𝜏/(1−𝛿)2(𝑝+2)}+‖‖𝑣_𝑡‖‖^{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)}≤𝑑‖𝑢‖^{2(𝑝+2)2(𝑝+2)}+𝐻(𝑡),‖𝑣‖^{2/(1−2𝛿)2(𝑝+2)}≤𝑑‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+𝐻(𝑡),∀𝑡≥0.(3.45) Therefore, (3.44) becomes _Ω𝑢𝑢_𝑡𝑑𝑥+_Ω𝑣𝑣_𝑡𝑑𝑥^1/(1−𝛿)≤𝐶‖𝑢‖^{2(𝑝+2)2(𝑝+2)}+‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²².(3.46) Note that Θ^1/(1−𝛿)(𝑡)=𝐻^1−𝛿(𝑡)+𝜖_Ω𝑢𝑢_𝑡+𝑣𝑣_𝑡𝑑𝑥^1/(1−𝛿)≤𝐶𝐻(𝑡)+||||_Ω𝑢𝑢_𝑡𝑑𝑥+_Ω𝑣𝑣_𝑡𝑑𝑥||||^1/(1−𝛿)≤𝐶𝐻(𝑡)+‖𝑢‖_{2(𝑝+2)2(𝑝+2)}+‖𝑣‖^{2(𝑝+2)2(𝑝+2)}+‖‖𝑢_𝑡‖‖²²+‖‖𝑣_𝑡‖‖²².(3.47) Combining (3.42) with (3.47), we have Θ(𝑡)≥𝐶Θ^1/(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.