Existence and Decay Estimate of Global Solutions to Systems of Nonlinear Wave Equations with Damping and Source Terms

Abstract

The initial-boundary value problem for a class of nonlinear wave equations system in bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set and obtain the asymptotic stability of global solutions through the use of a difference inequality.

1. Introduction

In this paper, we are concerned with the global solvability and decay stabilization for the following nonlinear wave equations system: 𝑢_𝑡𝑡−div|∇𝑢|^𝑝−2∇𝑢+𝑢_𝑡^𝑞−2𝑢_𝑡−Δ𝑢_𝑡=|𝑣|^𝑟+2|𝑢|^𝑟𝑢,(𝑥,𝑡)∈Ω×𝑅⁺,𝑣_𝑡𝑡−div|∇𝑣|^𝑝−2∇𝑣+𝑣_𝑡^𝑞−2𝑣_𝑡−Δ𝑣_𝑡=|𝑢|^𝑟+2|𝑣|^𝑟𝑣,(𝑥,𝑡)∈Ω×𝑅⁺ (1)(2) with the initial-boundary value conditions 𝑢(𝑥,0)=𝑢₀(𝑥)∈𝑊₀^1,𝑝(Ω),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥)∈𝐿²(Ω)𝑣()𝑥∈Ω,𝑥,0=𝑣₀(𝑥)∈𝑊₀^1,𝑝(Ω),𝑣_𝑡(𝑥,0)=𝑣₁(𝑥)∈𝐿²(Ω)𝑢()()()𝑥∈Ω,𝑥,𝑡=0,𝑣𝑥,𝑡=0,𝑥,𝑡∈𝜕Ω×𝑅⁺, (3)(4)(5) where Ω is a bounded open domain in 𝑅^𝑛 with a smooth boundary 𝜕Ω, 𝑝,𝑞≥2, 𝑟>0 and 𝑝<2(𝑟+2)≤𝑛𝑝/(𝑛−𝑝) for 𝑛≥𝑝 and 𝑝<2(𝑟+2)<+∞ for 𝑛<𝑝.

When 𝑝=2, Medeiros and Miranda [1] proved the existence and uniqueness of global weak solutions. Cavalcanti et al. in [2–4] considered the asymptotic behavior for wave equation and an analogous hyperbolic-parabolic system with boundary damping and boundary source term. In paper [5, 6], the authors dealt with the existence, uniform decay rates, and blowup for solutions of systems of nonlinear wave equations with damping and source terms.

Rammaha and Wilstein [7] and Yang [8] are concerned with the initial boundary value problem for a class of quasilinear evolution equations with nonlinear damping and source terms. Under appropriate conditions, by a Galerkin approximation scheme combined with the potential well method, they proved the existence and asymptotic behavior of global weak solutions when 𝑚<𝑝, where 𝑚≥0 and 𝑝 are, respectively, the growth orders of the nonlinear strain terms and the source term.

Ono [9] considers the following initial-boundary value problem for nonlinear wave equations with nonlinear dissipative terms: 𝑢_𝑡𝑡−Δ𝑢+𝛿₁𝑢_𝑡+𝛿₂𝑢_𝑡^𝛽𝑢_𝑡−𝛿₃Δ𝑢_𝑡=|𝑢|^𝛼()𝑢,𝑥,𝑡∈Ω×𝑅⁺,𝑢(𝑥,0)=𝑢₀(𝑥),𝑢_𝑡(𝑥,0)=𝑢₁(𝑥)𝑢(),𝑥∈Ω,𝑥,𝑡=0,𝑥∈𝜕Ω,𝑡≥0, (6) where 𝛿_𝑖≥0, 𝑖=1,2,3, and 𝛼,𝛽>0 are constants. The author mainly investigates on the blowup phenomenon to problem (6). On the other hand, in the case of 𝛿₁+𝛿₂+𝛿₃>0, he shows that the problem (6) admits a unique global solution, and its energy has some decay properties under some assumptions on 𝑢₀ and initial energy 𝐸(0)≡𝐸(𝑢₀,𝑢₁). In particular, when 𝛿₂>0 and 𝛿₁+𝛿₃>0 in (6), the energy 𝐸(𝑡)≡𝐸(𝑢(𝑡),𝑢_𝑡(𝑡)) has some polynomial and exponential decay rates, respectively.

For the following strongly damped nonlinear wave equation 𝑢_𝑡𝑡−Δ𝑢_𝑡−Δ𝑢+𝑓𝑢_𝑡+𝑔(𝑢)=ℎ, (7) Dell’Oro and Pata [10] obtain the long-time behavior of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space. In addition, the existence of global and local solutions, decay estimates, and blowup for solutions of nonlinear wave equation with source and damping terms and exponential nonlinearities are studied in [11–14].

In this paper, we prove the global existence for the problem (1)–(5) by applying the potential well theory introduced by Sattinger [15] and Payne and Sattinger [16]. Meanwhile, we obtain the asymptotic stabilization of global solutions by using a difference inequality [17].

For simplicity of notations, hereafter we denote by ‖⋅‖_𝑝 the norm of 𝐿^𝑝(Ω); ‖⋅‖ denotes 𝐿²(Ω) norm, and we write equivalent norm ‖⋅∇‖_𝑝 instead of 𝑊₀^1,𝑝(Ω) norm ‖⋅‖_𝑊0^1,𝑝_(Ω). Moreover, 𝐶 denotes various positive constants depending on the known constants and may be different at each appearance.

2. Local Existence

In this section, we investigate the local existence and uniqueness of the solutions of the problem (1)–(5). For this purpose, we list up two useful lemmas which will be used later and give the definition of weak solutions.

Lemma 1. Let 𝑢∈𝑊₀^1,𝑝(Ω), then 𝑢∈𝐿^𝑠(Ω); and the inequality ‖𝑢‖_𝑠≤𝐶‖𝑢‖_𝑊0^1,𝑝_(Ω) holds with a constant 𝐶>0 depending on Ω, 𝑝, and 𝑠, provided that 2≤𝑠<+∞, 2≤𝑛≤𝑝 and 2≤𝑠≤𝑛𝑝/(𝑛−𝑝), 2<𝑝<𝑛.

Lemma 2 (Young inequality). Let 𝑎,𝑏≥0 and 1/𝑝+1/𝑞=1 for 1<𝑝, 𝑞<+∞; then one has the inequality 𝑎𝑏≤𝛿𝑎^𝑝+𝐶(𝛿)𝑏^𝑞, (8) where 𝛿>0 is an arbitrary constant, and 𝐶(𝛿) is a positive constant depending on 𝛿.

Definition 3. A pair of functions (𝑢,𝑣) is said to be a weak solution of (1)–(5) on [0,𝑇] if 𝑢,𝑣∈𝐶([0,𝑇],𝑊₀^1,𝑝(Ω)), 𝑢_𝑡,𝑣_𝑡∈𝐶([0,𝑇],𝐿²(Ω)), [𝑢(0),𝑣(0)]=[𝑢₀,𝑣₀]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω), [𝑢_𝑡(0),𝑣_𝑡(0)]=[𝑢₁,𝑣₁]∈𝐿²(Ω)×𝐿²(Ω), and [𝑢,𝑣] satisfies 𝑢_𝑡(𝑡),𝜙_𝐿2(Ω)−𝑢₁,𝜙_𝐿2(Ω)+^𝑡₀|∇𝑢|^𝑝−2∇𝑢,∇𝜙_𝐿2(Ω)+𝑑𝜏_𝑡0𝑢_𝑡^𝑞−2𝑢_𝑡,𝜙_𝐿2(Ω)𝑑𝜏+^𝑡₀∇𝑢_𝑡,∇𝜙_𝐿2(Ω)=^𝑡₀|𝑣|^𝑟+2|𝑢|^𝑟𝑢,𝜙_𝐿2(Ω)𝑣𝑑𝜏,_𝑡(𝑡),𝜓_𝐿2(Ω)−𝑣₁,𝜓_𝐿2(Ω)+^𝑡₀|∇𝑣|^𝑝−2∇𝑣,∇𝜓_𝐿2(Ω)+𝑑𝜏_𝑡0𝑣_𝑡^𝑞−2𝑣_𝑡,𝜓_𝐿2(Ω)𝑑𝜏+^𝑡₀∇𝑣_𝑡,∇𝜓_𝐿2(Ω)=^𝑡₀|𝑢|^𝑟+2|𝑣|^𝑟𝑣,𝜓_𝐿2(Ω)𝑑𝜏, (9) for all test functions 𝜙,𝜓∈𝑊₀^1,𝑝(Ω) and for almost all 𝑡∈[0,𝑇].

The local existence and uniqueness of solutions for problem (1)–(5) can be proved through the use of Galerkin method. The result reads as follows.

Theorem 4 (local solution). Supposed that [𝑢₀,𝑣₀]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω), [𝑢₁,𝑣₁]∈𝐿²(Ω)×𝐿²(Ω), and 𝑝<2(𝑟+2)≤𝑛𝑝/(𝑛−𝑝) if 𝑛≥𝑝 and 𝑝<2(𝑟+2)<+∞ for 𝑛<𝑝, then there exists 𝑇>0 such that the problem (1)–(5) has a unique local solution [𝑢(𝑡),𝑣(𝑡)] satisfying [𝑢,𝑣]∈𝐿^∞[0,𝑇);𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω);𝑢_𝑡,𝑣_𝑡∈𝐿^∞[0,𝑇);𝐿²(Ω)×𝐿²(Ω,𝐸(𝑡)+_𝑡0)∇𝑢_𝜏(𝜏)²+∇𝑣_𝜏(𝜏)²+‖𝑢(𝜏)‖^𝑞_𝑞+‖𝑣(𝜏)‖^𝑞_𝑞𝑑𝜏=𝐸(0), (10)(11) where 𝐸(𝑡)=12𝑢_𝑡²+𝑣_𝑡²+1𝑝‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−1‖‖𝑟+2𝑢𝑣^𝑟+2_𝑟+2. (12)

Proof. Let {𝜔_𝑖}^∞_𝑖=1 be a basis for 𝑊₀^1,𝑝(Ω). Supposed that 𝑉_𝑘 is the subspace of 𝑊₀^1,𝑝(Ω) generated by {𝜔₁,𝜔₂,…,𝜔_𝑘},𝑘∈𝑁. We are going to look for the approximate solution 𝑢_𝑘(𝑡)=^𝑘_𝑖=1𝑔_𝑖𝑘(𝑡)𝜔_𝑖,𝑣_𝑘(𝑡)=^𝑘_𝑖=1ℎ_𝑖𝑘(𝑡)𝜔_𝑖 (13) which satisfies the following Cauchy problem: _Ω𝑢_𝑘^′′−div∇𝑢_𝑘^𝑝−2∇𝑢_𝑘+𝑢^′_𝑘^𝑞−2𝑢^′_𝑘−Δ𝑢^′_𝑘𝜔_𝑖=𝑑𝑥_Ω𝑣_𝑘^𝑟+2𝑢_𝑘^𝑟𝑢_𝑘𝜔_𝑖𝑑𝑥,_Ω𝑣_𝑘^′′−div∇𝑣_𝑘^𝑝−2∇𝑣_𝑘+𝑣^′_𝑘^𝑞−2𝑣^′_𝑘−Δ𝑣^′_𝑘𝜔_𝑖=𝑑𝑥_Ω𝑢_𝑘^𝑟+2𝑣_𝑘^𝑟𝑣_𝑘𝜔_𝑖𝑢𝑑𝑥,_𝑘(0)=𝑢_0𝑘=^𝑘_𝑖=1𝑢₀,𝜔_𝑖𝜔_𝑖⟶𝑢₀,in𝑊₀^1,𝑝(Ω),𝑣𝑘⟶∞,_𝑘(0)=𝑣_0𝑘=^𝑘_𝑖=1𝑣₀,𝜔_𝑖𝜔_𝑖⟶𝑣₀in𝑊₀^1,𝑝(Ω),𝑢𝑘⟶∞,_′𝑘(0)=𝑢_1𝑘=^𝑘_𝑖=1𝑢₁,𝜔_𝑖𝜔_𝑖⟶𝑢₁in𝐿²(Ω),𝑣𝑘⟶∞,_′𝑘(0)=𝑣_1𝑘=^𝑘_𝑖=1𝑣₁,𝜔_𝑖𝜔_𝑖⟶𝑣₁in𝐿²(Ω),𝑘⟶∞. (14)(15)(16)(17)(18)(19) Note that, we can solve the problem (14)–(19) by a Picard’s iteration method in ordinary differential equations. Hence, there exists a solution in [0,𝑇_𝑘) for some 𝑇_𝑘>0, and we can extend this solution to the whole interval [0,𝑇] for any given 𝑇>0 by making use of the a priori estimates below.
Multiplying (14) by 𝑔^′_𝑖𝑘(𝑡) and (15) by ℎ^′_𝑖𝑘(𝑡) and summing over 𝑖 from 1 to 𝑘, we obtain 12𝑑𝑢𝑑𝑡^′𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+𝑢^′_𝑘(𝑡)^𝑞_𝑞+∇𝑢^′_𝑘(𝑡)²=_Ω𝑣_𝑘^𝑟+2𝑢_𝑘^𝑟𝑢_𝑘𝑢^′_𝑘1𝑑𝑥,2𝑑𝑣𝑑𝑡^′𝑘(𝑡)²+∇𝑣_𝑘^𝑝_𝑝+𝑣^′_𝑘(𝑡)^𝑞_𝑞+∇𝑣^′_𝑘(𝑡)²=_Ω𝑢_𝑘^𝑟+2𝑣_𝑘^𝑟𝑣_𝑘𝑣^′_𝑘𝑑𝑥. (20)(21) By summing (20) and (21) and integrating the resulting identity over [0,𝑡], we have 12𝑢^′_𝑘(𝑡)²+𝑣^′_𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝+^𝑡₀∇𝑢^′_𝑘(𝑡)²+∇𝑣^′_𝑘(𝑡)²+𝑢^′_𝑘(𝜏)^𝑞_𝑞+𝑣^′_𝑘(𝜏)^𝑞_𝑞𝑑𝜏≤𝐶₀+^𝑡_0Ω𝑣_𝑘^𝑟+2𝑢_𝑘^𝑟𝑢_𝑘𝑢^′_𝑘+𝑢_𝑘^𝑟+2𝑣_𝑘^𝑟𝑣_𝑘𝑣^′_𝑘𝑑𝑥𝑑𝜏. (22) We estimate the right-hand terms of (22) as follows: we get from Hölder inequality and Lemmas 1 and 2 that ^𝑡_0Ω𝑣_𝑘^𝑟+2𝑢_𝑘^𝑟𝑢_𝑘𝑢^′_𝑘+𝑢_𝑘^𝑟+2𝑣_𝑘^𝑟𝑣_𝑘𝑣^′_𝑘≤𝑑𝑥𝑑𝜏_𝑡0𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²+𝑑𝜏_𝑡0Ω𝑢_𝑘𝑣_𝑘^2(𝑟+1)𝑢_𝑘²+𝑣_𝑘²𝑑𝑥𝑑𝜏≤^𝑡₀𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²𝑑𝜏+𝐶^𝑡₀𝑢_𝑘2()^2(𝑟+2)_𝑟+2+𝑣_𝑘2()^2(𝑟+2)_𝑟+2𝑑𝜏≤𝐶^𝑡₀𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²+∇𝑢_𝑘𝑝^2(𝑟+2)+∇𝑣_𝑘𝑝^2(𝑟+2)𝑑𝜏≤𝐶^𝑡₀𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝^{2(𝑟+2)/𝑝}𝑑𝜏. (23) It follows from (22) and (23) that 𝑢^′_𝑘(𝑡)²+𝑣^′_𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝+2^𝑡₀𝑢^′_𝑘(𝜏)^𝑞_𝑞‖+𝑣^′_𝑘(𝜏)^𝑞_𝑞+∇𝑢^′_𝑘(𝑡)²+∇𝑣^′_𝑘(𝑡)²𝑑𝜏≤2𝐶₀+𝐶^𝑡₀𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝^{2(𝑟+2)/𝑝}𝑑𝜏, (24) which implies that 𝑢^′_𝑘(𝑡)²+𝑣^′_𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝≤2𝐶₀+𝐶^𝑡₀𝑢^′_𝑘(𝜏)²+𝑣^′_𝑘(𝜏)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝^{2(𝑟+2)/𝑝}𝑑𝜏. (25) We get from (25) and Gronwall type inequality that 𝑢^′_𝑘(𝑡)²+𝑣^′_𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝≤2𝐶₀−2(𝑟+2)−𝑝𝑝𝐶𝑡^{−𝑝/(2(𝑟+2)−𝑝)}. (26) Thus, we deduce from (26) that there exists a time 𝑇>0 such that 𝑢^′_𝑘(𝑡)²+𝑣^′_𝑘(𝑡)²+∇𝑢_𝑘^𝑝_𝑝+∇𝑣_𝑘^𝑝_𝑝≤𝐶₁,∀𝑡∈[0,𝑇], (27) where 𝐶₁ is a positive constant independent of 𝑘.
We have from (24) and (26) that 2^𝑡₀𝑢^′_𝑘(𝜏)^𝑞_𝑞+𝑣^′_𝑘(𝜏)^𝑞_𝑞+∇𝑢^′_𝑘(𝜏)²+∇𝑣^′_𝑘(𝜏)²𝑑𝜏≤𝐶₂,∀𝑡∈[0,𝑇]. (28) It follows from (27) and (28) that 𝑢^′_𝑘(𝑡)²≤𝐶₁,𝑣^′_𝑘(𝑡)²≤𝐶₁,∇𝑢_𝑘^𝑝_𝑝≤𝐶₁,∇𝑣_𝑘^𝑝_𝑝≤𝐶₁.𝑢^′_𝑘(𝑡)and𝑣^′_𝑘(𝑡)areboundedin𝐿²[0,𝑇];𝐿^𝑞(Ω)and𝐿²[0,𝑇];𝐻¹₀(Ω). (29) Using the same process as the proof of Theorem 2.1 in paper [18], we derive that [𝑢(𝑡),𝑣(𝑡)] is a local solution of the problem (1)–(5). By (20) and (21), we conclude that (11) is valid.

3. Global Existence

In order to state our main results, we first introduce the following functionals: 𝐽=1([𝑢,𝑣])𝑝‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−1‖‖𝑟+2𝑢𝑣^𝑟+2_𝑟+2,𝐾([𝑢,𝑣])=‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−2‖𝑢𝑣‖^𝑟+2_𝑟+2 (30)(31) for [𝑢,𝑣]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω).

We put that 𝑑=infsup_𝜆≥0𝐽(𝜆[𝑢,𝑣])∶[𝑢,𝑣]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω)/{[0,0]}. (32) Then, we are able to define the stable set as follows for problem (1)–(5): 𝑊=[𝑢,𝑣]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω)𝐽∪.∣𝐾([𝑢,𝑣])>0,([𝑢,𝑣])<𝑑{[0,0]} (33) We denote the total energy related to (1) and (2) by (12), and 𝐸(0)=12𝑢₁²+𝑣₁²+1𝑝∇𝑢₀^𝑝_𝑝+∇𝑣₀^𝑝_𝑝−1𝑢𝑟+2⁰𝑣₀^𝑟+2_𝑟+2 (34) is the total energy of the initial data.

Lemma 5. Let [𝑢,𝑣] be a solution to problem (1)–(5); then, 𝐸(𝑡) is a nonincreasing function for 𝑡>0 and 𝑑𝐸(𝑡)𝑢𝑑𝑡=−_𝑡^𝑞_𝑞+𝑣_𝑡^𝑞_𝑞+∇𝑢_𝑡²₂+∇𝑣_𝑡²₂. (35)

We have from (11) that 𝐸(𝑡) is the primitive of an integrable function. Therefore, 𝐸(𝑡) is absolutely continuous, and equality (35) is satisfied.

Lemma 6. Supposed that [𝑢,𝑣]∈𝑊₀^1,𝑝(Ω)×𝑊₀^1,𝑝(Ω), and 𝑝<2(𝑟+2)≤𝑛𝑝/(𝑛−𝑝) if 𝑛≥𝑝; 𝑝<2(𝑟+2)<+∞ if 𝑛<𝑝, then 𝑑>0.

Proof. Since 𝐽(𝜆[=𝜆𝑢,𝑣])^𝑝𝑝‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−𝜆^2(𝑟+2)‖‖𝑟+2𝑢𝑣^𝑟+2_𝑟+2, (36) so we get 𝑑𝐽(𝜆[𝑑𝜆𝑢,𝑣])=𝜆^𝑝−1‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−2𝜆^2𝑟+3‖𝑢𝑣‖^𝑟+2_𝑟+2. (37) In case 𝑢𝑣≠0, let (𝑑/𝑑𝜆)𝐽(𝜆[𝑢,𝑣])=0, which implies that 𝜆₁=‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝2‖𝑢𝑣‖^𝑟+2_𝑟+2^{1/(2𝑟−𝑝+4)}. (38)
As 𝜆=𝜆₁, an elementary calculation shows that (𝑑²/𝑑𝜆²)𝐽(𝜆[𝑢,𝑣])|_𝜆=𝜆1<0. Therefore, we have that sup_𝜆≥0𝐽(𝜆[𝜆𝑢,𝑣])=𝐽₁[]=2()𝑢,𝑣𝑟+2−𝑝()‖‖2𝑝𝑟+2∇𝑢^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝2^{𝑝/(2𝑟+4)}‖𝑢𝑣‖^𝑝/2_𝑟+2^{(2𝑟+4)/(2𝑟−𝑝+4)}. (39) It follows from Hölder inequality and Lemma 1 that ‖𝑢𝑣‖^𝑝/2_𝑟+2≤‖𝑢‖₂₍₎^𝑝/2_𝑟+2‖𝑣‖₂₍₎^𝑝/2_𝑟+2≤12‖𝑢‖^𝑝_2()𝑟+2+‖𝑣‖^𝑝_2()𝑟+2≤𝐶‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝. (40)
We get from (39) and (40) that sup_𝜆≥0𝐽(𝜆[≥2()𝑢,𝑣])𝑟+2−𝑝()22𝑝𝑟+2^{𝑝/(2𝑟+4)}𝐶^{−(2𝑟+4)/(2𝑟−𝑝+4)}>0. (41) In case 𝑢𝑣=0 and 𝑢=0 or 𝑣=0, then 𝐽(𝜆[=𝜆𝑢,𝑣])^𝑝𝑝‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝. (42) Therefore, we have 𝐽(𝜆[𝑢,𝑣])=+∞. (43) We conclude from (41) and (43) that 2()𝑑≥𝑟+2−𝑝()22𝑝𝑟+2^{𝑝/(2𝑟+4)}𝐶^{−(2𝑟+4)/(2𝑟−𝑝+4)}>0. (44) Thus, we complete the proof of Lemma 6.

Lemma 7. Supposed that 𝑝<2(𝑟+2)≤𝑛𝑝/(𝑛−𝑝) for 𝑛≥𝑝 and 𝑝<2(𝑟+2)<+∞ for 𝑛<𝑝, if [𝑢₀,𝑣₀]∈𝑊, [𝑢₁,𝑣₁]∈𝐿²(Ω)×𝐿²(Ω) and 𝐸(0)<𝑑, then [𝑢,𝑣]∈𝑊 for ∀𝑡∈[0,𝑇).

Proof. Assume that there exists a number 𝑡^∗∈(0,𝑇) such that [𝑢(𝑡),𝑣(𝑡)]∈𝑊 on [0,𝑡^∗) and 𝑢(𝑡^∗)∉𝑊. Then, in virtue of the continuity of 𝑢(𝑡), we see 𝑢(𝑡^∗)∈𝜕𝑊, where 𝜕𝑊 denotes the boundary of domain 𝑊. From the definition of 𝑊 and the continuity of 𝐽([𝑢(𝑡),𝑣(𝑡)]) and 𝐾([𝑢(𝑡),𝑣(𝑡)]) in 𝑡, we have either 𝐽𝑢𝑡^∗,𝑣𝑡^∗=𝑑 (45) or 𝐾𝑢𝑡^∗,𝑣𝑡^∗=0. (46) It follows from (12) and (30) that 𝐽𝑢𝑡^∗,𝑣𝑡^∗≤𝐸𝑡^∗≤𝐸(0)<𝑑. (47) So, case (45) is impossible.
Assume that (46) holds; then, we get that 𝑑𝐽𝜆𝑢𝑡𝑑𝜆^∗,𝑣𝑡^∗=𝜆^𝑝−11−𝜆^{2𝑟−𝑝+4}‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝. (48) We obtain from (𝑑/𝑑𝜆)𝐽(𝜆[𝑢(𝑡^∗),𝑣(𝑡^∗)])=0 that 𝜆=1. Since 𝑑²𝑑𝜆²𝐽𝜆𝑢𝑡^∗,𝑣𝑡^∗_𝜆=1=−2(𝑟+2)−𝑝+(2𝑟+3)<0. (49) Consequently, we get from (47) that sup_𝜆≥0𝐽𝜆𝑢𝑡^∗,𝑣𝑡^∗=𝐽𝑢𝑡^∗,𝑣𝑡^∗<𝑑, (50) which contradicts the definition of 𝑑. Hence, case (46) is impossible as well. Thus we conclude that [𝑢(𝑡),𝑣(𝑡)]∈𝑊 on [0,𝑇).

Theorem 8 (global solution). Supposed that 𝑝<2(𝑟+2)≤𝑛𝑝/(𝑛−𝑝) as 𝑛≥𝑝 and 𝑝<2(𝑟+2)<+∞ as 𝑛<𝑝, and [𝑢(𝑡),𝑣(𝑡)] is a local solution of problem (1)–(5) on [0,𝑇). If [𝑢₀,𝑣₀]∈𝑊, [𝑢₁,𝑣₁]∈𝐿²(Ω)×𝐿²(Ω) and 𝐸(0)<𝑑, then [𝑢(𝑡),𝑣(𝑡)] is a global solution of problem (1)–(5).

Proof. It suffices to show that ‖𝑢_𝑡‖²+‖𝑣_𝑡‖²+‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝 is bounded uniformly with respect to 𝑡. Under the hypotheses in Theorem 8, we get from Lemma 7 that [𝑢,𝑣]∈𝑊 on [0,𝑇). So the following formula holds on [0,𝑇): 𝐽=1([𝑢,𝑣])𝑝‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝−1‖‖𝑟+2𝑢𝑣^𝑟+2_𝑟+2≥2(𝑟+2)−𝑝()‖‖2𝑝𝑟+2∇𝑢^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝. (51) We have from (51) that 12𝑢_𝑡²+𝑣_𝑡²+2(𝑟+2)−𝑝()‖‖2𝑝𝑟+2∇𝑢^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤12𝑢_𝑡²+𝑣_𝑡²𝑢(𝑡)(𝑡(𝑡)(0)+𝐽([,𝑣)])=𝐸≤𝐸<𝑑. (52) Hence, we get 𝑢_𝑡²+𝑣_𝑡²+‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝()≤max2,2𝑝𝑟+22(𝑟+2)−𝑝𝑑<+∞. (53) The above inequality and the continuation principle lead to the global existence of the solution [𝑢,𝑣] for problem (1)–(5).

4. Asymptotic Behavior of Global Solutions

The following lemma plays an important role in studying the decay estimate of global solutions for the problem (1)–(5).

Lemma 9 (see [9]). Suppose that 𝜑(𝑡) is a nonincreasing nonnegative function on [0,+∞) and satisfies 𝜑(𝑡)^𝑟+1≤𝑘𝜑(𝑡)−𝜑(𝑡+1),∀𝑡≥0. (54) Then, 𝜑(𝑡) has the decay property 𝜑(𝑡)≤𝑟𝑘(𝑡−1)+𝑀^{−𝑟−1/𝑟},∀𝑡≥1, (55) where 𝑘,𝑟>0 are constants and 𝑀=max_𝑡∈[0,1]𝜑(𝑡).

Lemma 10. Under the assumptions of Theorem 8, if initial value [𝑢₀,𝑣₀]∈𝑊 and [𝑢₁,𝑣₁]∈𝐿²(Ω)×𝐿²(Ω) are sufficiently small such that 𝐶^2(𝑟+2)2𝑝(𝑟+2)()𝐸(0)2𝑝𝑟+2−𝑝^{(2(𝑟+2)−𝑝)/𝑝}<1, (56) then ‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤1𝜃𝐾([𝑢,𝑣]), (57) where 𝜃=1−𝐶^2(𝑟+2)((2𝑝(𝑟+2)/(2𝑝(𝑟+2)−𝑝))𝐸(0))^{(2(𝑟+2)−𝑝)/𝑝}>0 is a positive constant and 𝐶 is the optimal Sobolev’s constant from 𝑊₀^1,𝑝(Ω) to 𝐿^2(𝑟+2)(Ω).

Proof. We have from Lemma 1 and (52) that 2‖𝑢𝑣‖^𝑟+2_𝑟+2≤2‖𝑢‖₂₍₎^𝑟+2_𝑟+2‖𝑣‖₂₍₎^𝑟+2_𝑟+2≤‖𝑢‖₂₍₎^2(𝑟+2)_𝑟+2+‖𝑣‖₂₍₎^2(𝑟+2)_𝑟+2≤𝐶^2(𝑟+2)‖∇𝑢‖_𝑝^2(𝑟+2)+‖∇𝑣‖_𝑝^2(𝑟+2)≤𝐶^2(𝑟+2)‖∇𝑢‖_𝑝^{2(𝑟+2)−𝑝}‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖_𝑝^{2(𝑟+2)−𝑝}‖∇𝑣‖^𝑝_𝑝≤𝐶^2(𝑟+2)2𝑝(𝑟+2)()𝐸(0)2𝑝𝑟+2−𝑝^{(2(𝑟+2)−𝑝)/𝑝}×‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝. (58) Therefore, we get from (58) and (31) that 1−𝐶^2(𝑟+2)2𝑝(𝑟+2)()𝐸(0)2𝑝𝑟+2−𝑝^{(2(𝑟+2)−𝑝)/𝑝}×‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤𝐾([𝑢,𝑣]). (59) Let 𝜃=1−𝐶^2(𝑟+2)2𝑝(𝑟+2)()𝐸(0)2𝑝𝑟+2−𝑝^{(2(𝑟+2)−𝑝)/𝑝}>0; (60) then, we have from (59) that ‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤1𝜃𝐾([𝑢,𝑣]). (61)

Theorem 11. Under the assumptions of Theorem 8, if 𝑝<𝑞<𝑟+2 and (56) hold, then the global solution [𝑢,𝑣] in 𝑊 of the problem (1)–(5) has the following decay property: 𝐸(𝑡)≤𝑝−2()𝑝𝐶𝑡−1+𝑀^{(𝑝+𝑞−𝑝𝑞)/𝑝𝑝/(𝑝+𝑞−𝑝𝑞)},∀𝑡>1, (62) where 𝑀=max_𝑡∈[0,1]𝐸(𝑡)>0 is some constant depending only on [𝑢₀,𝑣₀] and [𝑢₁,𝑣₁].

Proof. Multiplying (1) by 𝑢_𝑡 and (2) by 𝑣_𝑡 and integrating over Ω×[𝑡,𝑡+1], and summing up together, we get _𝑡^𝑡+1𝑢_𝑡(𝑠)^𝑞_𝑞+𝑣_𝑡(𝑠)^𝑞_𝑞+∇𝑢_𝑡(𝑠)²₂+∇𝑣_𝑡(𝑠)²₂𝑑𝑠=𝐸(𝑡)−𝐸(𝑡+1). (63) Thus, there exists 𝑡₁∈[𝑡,𝑡+1/4], 𝑡₂∈[𝑡+3/4,𝑡+1] such that 4𝑢_𝑡𝑡_𝑖^𝑞_𝑞+𝑣_𝑡𝑡_𝑖^𝑞_𝑞+∇𝑢_𝑡𝑡_𝑖²₂+∇𝑣_𝑡𝑡_𝑖²₂=𝐸(𝑡)−𝐸(𝑡+1),𝑖=1,2. (64)
On the other hand, we multiply (1) by 𝑢 and (2) by 𝑣 and integrate over Ω×[𝑡₁,𝑡₂]. Adding them together, we obtain ^𝑡2_𝑡1𝐾([𝑢,𝑣])𝑑𝑠=^𝑡2_𝑡1𝑢_𝑡²𝑑𝑠+^𝑡2_𝑡1𝑣_𝑡²+𝑢𝑑𝑠_𝑡𝑡₁,𝑢𝑡₁−𝑢_𝑡𝑡₂,𝑢𝑡₂+𝑣_𝑡𝑡₁,𝑣𝑡₂−𝑣_𝑡𝑡₂𝑣𝑡₂−^𝑡2_𝑡1Ω𝑢_𝑡^𝑞−2𝑢_𝑡+𝑢𝑑𝑥𝑑𝑠_𝑡2𝑡1Ω𝑣_𝑡^𝑞−2𝑣_𝑡−𝑣𝑑𝑥𝑑𝑠_𝑡2𝑡1Ω∇𝑢_𝑡∇𝑢𝑑𝑥𝑑𝑠−^𝑡2_𝑡1Ω∇𝑣_𝑡∇𝑣𝑑𝑥𝑑𝑠. (65) From (63), Sobolev inequality, and Hölder inequality, we have ^𝑡2_𝑡1𝑢_𝑡²𝑑𝑠≤𝐶^𝑡2_𝑡1∇𝑢_𝑡²𝑑𝑠≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))_𝑡2𝑡1,𝑣_𝑡²𝑑𝑠≤𝐶^𝑡2_𝑡1∇𝑣_𝑡²𝑑𝑠≤𝐶(𝐸(𝑡)−𝐸(𝑡+1)). (66) We get from (52), (64), and Lemmas 1 and 2 that 𝑢_𝑡𝑡_𝑖,𝑢𝑡_𝑖≤𝑢_𝑡𝑡_𝑖⋅𝑢𝑡_𝑖≤𝐶∇𝑢_𝑡𝑡_𝑖⋅∇𝑢(𝑡_𝑖)_𝑝≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)^1/𝑝≤𝐶(𝜀)(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)𝑣,𝑖=1,2,_𝑡𝑡_𝑖,𝑣𝑡_𝑖≤𝑣_𝑡𝑡_𝑖⋅𝑣𝑡_𝑖≤𝐶∇𝑣_𝑡𝑡_𝑖⋅∇𝑣(𝑡_𝑖)_𝑝≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)^1/𝑝≤𝐶(𝜀)(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠),𝑖=1,2. (67) From Hölder inequality and Lemma 2,we get ^𝑡2_𝑡1Ω𝑢_𝑡^𝑞−2𝑢_𝑡𝑢𝑑𝑥𝑑𝑠≤^𝑡2_𝑡1𝑢_𝑡𝑞^𝑞−1‖𝑢‖_𝑞≤𝑑𝑠_𝑡2𝑡1𝑢_𝑡^𝑞_𝑞𝑑𝑠^{(𝑞−1)/𝑞𝑡2}_𝑡1‖𝑢‖^𝑞_𝑞𝑑𝑠^1/𝑞≤𝐶(𝜀)^𝑡2_𝑡1𝑢_𝑡^𝑞_𝑞𝑑𝑠+𝜀^𝑡2_𝑡1‖𝑢‖^𝑞_𝑞𝑑𝑠,_𝑡2𝑡1Ω𝑣_𝑡^𝑞−2𝑣_𝑡𝑣𝑑𝑥𝑑𝑠≤^𝑡2_𝑡1𝑣_𝑡𝑞^𝑞−1‖𝑣‖_𝑞≤𝑑𝑠_𝑡2𝑡1𝑣_𝑡^𝑞_𝑞𝑑𝑠^{(𝑞−1)/𝑞𝑡2}_𝑡1‖𝑣‖^𝑞_𝑞𝑑𝑠^1/𝑞≤𝐶(𝜀)^𝑡2_𝑡1𝑣_𝑡^𝑞_𝑞𝑑𝑠+𝜀^𝑡2_𝑡1‖𝑣‖^𝑞_𝑞𝑑𝑠. (68)(69)
Since 𝑝<𝑞<𝑟+2 and the property of the function 𝑓(𝑥)=𝛼^𝑥/𝑥, 𝛼≥0, 𝑥>0, we obtain ‖𝑢‖^𝑞_𝑞𝑞‖𝑢‖≤𝐶^𝑝𝑝𝑝‖𝑢‖+𝐶^𝑟+2𝑟+2,‖𝑣‖𝑟+2𝑞^𝑞𝑞‖𝑣‖≤𝐶^𝑝𝑝𝑝‖𝑣‖+𝐶^𝑟+2𝑟+2.𝑟+2 (70)
We conclude from (69), (70), [𝑢,𝑣]∈𝑊, and Lemma 1 that ‖𝑢‖^𝑞_𝑞+‖𝑣‖^𝑞_𝑞≤𝐶‖𝑢‖^𝑝_𝑝+‖𝑢‖^𝑟+2_𝑟+2+‖𝑣‖^𝑝_𝑝+‖𝑣‖^𝑟+2_𝑟+2≤𝐶‖𝑢‖^𝑝_𝑝+‖∇𝑢‖^𝑝_𝑝+‖𝑣‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤𝐶‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝≤𝐶𝐸(𝑡). (71) It follows from (63), (68), (69), and (71) that −^𝑡2_𝑡1Ω𝑢_𝑡^𝑞−2𝑢_𝑡𝑢𝑑𝑥𝑑𝑠+^𝑡2_𝑡1Ω𝑣_𝑡^𝑞−2𝑣_𝑡(𝜀𝐸(𝑡)(𝑣𝑑𝑥𝑑𝑠≤𝐶)(−𝐸𝑡+1))+𝜀𝐶^𝑡2_𝑡1𝐸(𝑠)𝑑𝑠, (72) and we obtain from (63), Sobolev inequality, Hölder inequality, and Lemma 2 that −^𝑡2_𝑡1Ω∇𝑢_𝑡∇𝑢𝑑𝑠≤^𝑡2_𝑡1∇𝑢_𝑡⋅‖‖≤∇𝑢𝑑𝑠_𝑡2𝑡1∇𝑢_𝑡²𝑑𝑠^1/2𝑡2_𝑡1‖∇𝑢‖²𝑑𝑠^1/2≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2𝑡2_𝑡1‖∇𝑢‖²_𝑝𝑑𝑠^1/2≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2𝑡2_𝑡1‖∇𝑢‖^𝑝_𝑝𝑑𝑠^1/𝑝≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}+𝜀^𝑡2_𝑡1‖∇𝑢‖^𝑝_𝑝𝑑𝑠. (73) Similarly, we have the following formula: −^𝑡2_𝑡1Ω∇𝑣_𝑡∇𝑣𝑑𝑠≤^𝑡2_𝑡1∇𝑣_𝑡⋅‖‖≤∇𝑣𝑑𝑠_𝑡2𝑡1∇𝑣_𝑡²𝑑𝑠^1/2𝑡2_𝑡1‖∇𝑣‖²𝑑𝑠^1/2≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2𝑡2_𝑡1‖∇𝑣‖²_𝑝𝑑𝑠^1/2≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^1/2𝑡2_𝑡1‖∇𝑣‖^𝑝_𝑝𝑑𝑠^1/𝑝≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}+𝜀^𝑡2_𝑡1‖∇𝑣‖^𝑝_𝑝𝑑𝑠. (74) We get from (57), (73), and (74) that ^𝑡2_𝑡1Ω∇𝑢_𝑡∇𝑢+∇𝑣_𝑡(𝐸(𝑡)(∇𝑣𝑑𝑠≤𝐶−𝐸𝑡+1))^{𝑝/2(𝑝−1)}+𝜀^𝑡2_𝑡1‖∇𝑢‖^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝(𝐸(𝑡)(𝑑𝑠≤𝐶−𝐸𝑡+1))^{𝑝/2(𝑝−1)}+𝜀𝜃^𝑡2_𝑡1𝐾([𝑢,𝑣])𝑑𝑠. (75)
Choosing small enough 𝜀, we have from (65), (66), (67), (72), and (75) that ^𝑡2_𝑡1𝐾𝐸(𝑡)(+(𝐸(𝑡)(([𝑢,𝑣])𝑑𝑠≤𝐶(−𝐸𝑡+1))−𝐸𝑡+1))^{𝑝/2(𝑝−1)}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)+𝜀^𝑡2_𝑡1𝐸(𝑠)𝑑𝑠. (76) It follows from (30) and (31) that 𝐽=2()([𝑢,𝑣])𝑟+2−𝑝()‖‖2𝑝𝑟+2∇𝑢^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝+12()𝐾.𝑟+2([𝑢,𝑣]) (77) On the other hand, from (12) and using (57) and (77), we deduce that 𝐸(𝑡)=12𝑢_𝑡²+𝑣_𝑡²=1+𝐽([𝑢,𝑣])2𝑢_𝑡²+𝑣_𝑡²+2(𝑟+2)−𝑝()×‖‖2𝑝𝑟+2∇𝑢^𝑝_𝑝+‖∇𝑣‖^𝑝_𝑝+12()𝐾≤1𝑟+2([𝑢,𝑣])2𝑢_𝑡²+𝑣_𝑡²+2(𝑟+2)−𝑝()+12𝜃𝑝𝑟+22().𝑟+2×𝐾([𝑢,𝑣]) (78) By integrating (78) over [𝑡₁,𝑡₂], we obtain ^𝑡2_𝑡1𝐸(𝑠)1𝑑𝑠≤2^𝑡2_𝑡1𝑢_𝑡²+𝑣_𝑡²+2()𝑑𝑠𝑟+2−𝑝()+12𝜃𝑝𝑟+22(𝑟+2)^𝑡2_𝑡1𝐾([𝑢,𝑣])𝑑𝑠. (79) For small enough 𝜀, we have from (76) and (79) that ^𝑡2_𝑡1𝐸(𝑠)𝐸(𝑡)(+(𝐸(𝑡)(𝑑𝑠≤𝐶(−𝐸𝑡+1))−𝐸𝑡+1))^{𝑝/(2(𝑝−1))}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠). (80) Thus, there exists 𝑡^∗∈[𝑡₁,𝑡₂], such that 𝐸𝑡^∗≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))+(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠). (81) Multiplying (1) by 𝑢_𝑡 and (2) by 𝑣_𝑡 and integrating over Ω×[𝑡^∗,𝑡₂], and summing up, we get 𝐸𝑡₂=𝐸𝑡^∗−^𝑡2_𝑡∗𝑢_𝑡^𝑞_𝑞+𝑣_𝑡^𝑞_𝑞+∇𝑢_𝑡²+∇𝑣_𝑡²𝑑𝑠. (82) Therefore, we obtain from (63), (81), and (82) that sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)𝐸(𝑡)(+(𝐸(𝑡)(≤𝐶(−𝐸𝑡+1))−𝐸𝑡+1))^{𝑝/2(𝑝−1)}+𝜀sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠). (83) Choosing small enough 𝜀, we have from (83) that sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)𝐸(𝑡)(+(𝐸(𝑡)(≤𝐶(−𝐸𝑡+1))−𝐸𝑡+1))^{𝑝/2(𝑝−1)}. (84)
Since 𝑝>2 and 𝐸(𝑡)<𝐸(0), we get sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)≤𝐶(𝐸(𝑡)−𝐸(𝑡+1))^{𝑝/2(𝑝−1)}. (85) Consequently, sup_{𝑡≤𝑠≤𝑡+1}𝐸(𝑠)^{(2(𝑝−1))/𝑝}≤𝐶(𝐸(𝑡)−𝐸(𝑡+1)). (86) Thus, applying Lemma 9 to (86), we get 𝐸(𝑡)≤𝑝−2()𝑝𝐶𝑡−1+𝑀^{(𝑝−2)/𝑝𝑝/(2−𝑝)},∀𝑡>1, (87) where 𝑀=max_𝑡∈[0,1]𝐸(𝑡)>0 is some constant depending only on [𝑢₀,𝑣₀] and [𝑢₁,𝑣₁].

Acknowledgments

This research was supported by the National Natural Science Foundation of China (no. 61273016), The Natural Science Foundation of Zhejiang Province (no. Y6100016), The Middle-aged and Young Leader in Zhejiang University of Science and Technology (2008–2012), and the Interdisciplinary Pre-research Project of Zhejiang University of Science and Technology (2010–2012).