Weakly Compact Uniform Attractor for the Nonautonomous Long-Short Wave Equations

Abstract

Solutions and weakly compact uniform attractor for the nonautonomous long-short wave equations with translation compact forces were studied in a bounded domain. We first established the existence and the uniqueness of the solution to the system by using Galerkin method and then obtained the uniform absorbing set and the weakly compact uniform attractor of the problem by applying techniques of constructing skew product flow in the extended phase space.

1. Introduction

The long wave-short wave (LS) resonance equations arise in the study of the interaction of the surface waves with both gravity and capillary modes presence and also in the analysis of internal waves, as well as Rossby wave [1]. In the plasma physics they describe the resonance of the high-frequency electron plasma oscillation and the associated low-frequency ion density perturbation [2]. Benney [3] presents a general theory for the interaction between the short wave and the long wave.

Due to their rich physical and mathematical properties the long wave-short wave resonance equations have drawn much attention of many physicists and mathematicians. For one-dimensional propagation of waves, there are many studies on this interaction. Guo [4, 5] obtains the existence of global solution for long-short wave equations and generalized long-short wave equations, respectively. The existence of global attractor was studied in [6–8]. The orbital stability of solitary waves for this system has been studied in [9]. In [10], Guo investigated the asymptotic behavior of solutions for the long-short wave equations with zero order dissipation in 𝐻²_per×𝐻¹_per. The approximation inertial manifolds for LS type equations have been studied in [11]. The well posedness of the Cauchy problem for the long wave-short wave resonance equations was studied in [8, 12–17].

In this paper, nonautonomous LS equations with translation compact forces were studied. The essential difference between nonautonomous systems and autonomous ones is that the former get much influenced by the time-depended external forces, which breaks semigroup property of the flow or semiflow created by autonomous systems. Also, attractors of nonautonomous systems are no longer invariable; they change with the changing of the initial time. This makes it impossible for us to consider nonautonomous systems completely in the same way of autonomous ones. Fortunately, Chepyzhov and Vishik [18, 19] developed techniques by which skills in the study of autonomous systems can be used in dealing with nonautonomous problems. Their central idea is that constructing skew product flow in extended phase space is obtained by 𝑆(𝑡)(𝑢,𝜎)=𝑈_𝜎(𝑡,0)𝑢,𝑇(𝑡)𝜎,𝑡≥0,(𝑢,𝜎)∈𝐸×, (1) where {𝑈_𝜎∈∑(𝑡,𝜏)} is a family of processes, {𝑇(𝑡)} is a translation semigroup, and the flow {𝑆(𝑡)} can be proved to be a semigroup under some preconditions, such as the translation identity and (𝐸×∑,𝐸)-continuity of {𝑈_𝜎(𝑡,𝜏)}, and more importantly, the compactness of the symbol space ∑. By this means, we can get the uniform attractor by projecting the global attractor of {𝑆(𝑡)} to the phase space if the latter exists. We consider the following nonautonomous dissipative generalized long-short wave equations: 𝑖𝑢_𝑡+𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²()𝑛𝑢=𝑎𝑥,𝑡,_𝑡+𝛽𝑛+|𝑢|²_𝑥+𝑓|𝑢|²=𝑏(𝑥,𝑡), (2)(3) with the initial conditions 𝑢(𝑥,𝜏)=𝑢_𝜏(𝑥),𝑛(𝑥,𝜏)=𝑛_𝜏(𝑥), (4) and the boundary value conditions 𝑢(𝑥,𝑡)∣_𝜕Ω=0,𝑛(𝑥,𝑡)∣_𝜕Ω=0, (5) where 𝑥∈Ω=(−𝐷,𝐷)⊂ℝ. 𝑡≥𝜏∈ℝ₊. Nonautonomous terms 𝑎(𝑥,𝑡) and 𝑏(𝑥,𝑡) are time-depended external forces, which are supposed to be translation compact (cf. [18] or Assumption 1). Nonlinear terms 𝑓(𝑠) and 𝑔(𝑠) are given smooth and real, satisfying 𝑓(𝑠)⩽𝑐₁𝑠^𝑝/2+1,𝑔(𝑠)⩽𝑐₂𝑠^1/2,𝑓+1_(𝑘)(𝑠)≤𝑐₃,𝑔^(𝑘)(𝑠)≤𝑐₄, (6)(7) where 0≤𝑠<∞, 𝑝<2, 𝑘=1,2, and constants 𝛼, 𝛽, and 𝑐_𝑗 are given in ℝ₊ for 𝑗=1,2,3,4.

Our aim here is, firstly, to get the unique existence of solutions for problem (2)~(5) and then to derive the existence of weakly compact uniform attractor for it with the above-mentioned method. Here, and throughout this paper, uniform means uniform about symbols (𝜎) in symbol space (∑) unless there is special explanation. In fact, it is the same if we say uniform about the initial time, since the translation identity and the (𝐸×∑,𝐸)-continuity of {𝑈_𝜎(𝑡,𝜏)} hold in our case (cf. [20]).

Throughout this paper, we denote by ‖⋅‖ the norm of 𝐻=𝐿²(Ω) with usual inner product (⋅,⋅), denote by ‖⋅‖ the norm of 𝐿^𝑝(Ω) for all 1≤𝑝≤∞(‖⋅‖₂=‖⋅‖), and denote by ‖⋅‖_𝐻^𝑘 the norm of a usual Sobolev space 𝐻^𝑘(Ω) for all 1≤𝑘≤∞. And we denote different constants by a same letter 𝐶, and 𝐶(⋅,⋅) represents that the constant relies only on the parameters appearing in the brackets.

This paper is organized as follows. In Section 2, we recall some facts about the nonautonomous system. In Section 3, we provide the uniform a priori estimates in time. In Section 4, we obtain the unique existence of the solutions for problem (2)~(5) by Galerkin method. Section 5 contains the weakly compact uniform attractor for the nonautonomous system (2)~(5), and in the proof of Theorem 13, the (𝐸₀×∑,𝐸₀)-continuity of {𝑈_𝜎∈∑(𝑡,𝜏)} is proved.

2. Preliminary Results

Let ℑ be a topological space, and let 𝜑(𝑠)∈ℑ be a function. The set ℋ𝜑={𝜑(ℎ+𝑠)∣ℎ∈ℝ} (8) is called the hull of 𝜑 in ℑ, denoted by ℋ(𝜑). 𝜑 is translation compact if ℋ(𝜑) is compact in ℑ.

We denote all the translation compact functions in 𝐿²_loc(ℝ;𝑋) by 𝐿²_𝑐(ℝ;𝑋), where 𝑋 is a Banach space. Apparently, 𝜑∈𝐿²_𝑐(ℝ;𝑋) implies that 𝜑 is translation bounded; that is, 𝜑²_{𝐿2𝑏(ℝ;𝑋)}=sup_{𝑡∈ℝ𝑡}^𝑡+1𝜑²_𝑋𝑑𝑠<∞. (9)

Let 𝐸 be a Banach space, and let a family of two-parameter mappings {𝑈(𝑡,𝜏)}={𝑈(𝑡,𝜏)∣𝑡≥𝜏,𝜏∈ℝ} act in 𝐸. We also need the following definitions and lemma (cf. [19, 20]).

Definition 1. Let ∑ be a parameter set. {𝑈_𝜎(𝑡,𝜏),𝑡≥𝜏,𝜏∈ℝ}, 𝜎∈∑ is said to be a family of processes in Banach space 𝐸, if for each 𝜎∈∑, {𝑈_𝜎(𝑡,𝜏)} from 𝐸 to 𝐸 satisfies 𝑈_𝜎(𝑡,𝑠)∘𝑈_𝜎(𝑠,𝜏)=𝑈_𝜎()𝑈𝑡,𝜏,∀𝑡≥𝑠≥𝜏,𝜏∈ℝ,_𝜎(𝜏,𝜏)=𝐼,𝐼istheidentityoperator,𝜏∈ℝ. (10)

Definition 2. {𝑈_𝜎∈∑(𝑡,𝜏)}, a family of processes in Banach space 𝐸, is called (𝐸×∑,𝐸)-continuous, if for all fixed 𝑇 and 𝜏, 𝑇≥𝜏,  projection (𝑢_𝜏,𝜎)→𝑈_𝜎(𝑇,𝜏)𝑢_𝜏  is continuous from 𝐸×∑ to 𝐸.

A set 𝐵₀⊂𝐸 is said to be uniformly absorbing set for the family of processes {𝑈_𝜎∈∑(𝑡,𝜏)}, if for any 𝜏∈ℝ and 𝐵∈ℬ(𝐸) which denotes the set of all bounded subsets of 𝐸, there exists 𝑡₀=𝑡₀(𝜏,𝐵)≥𝜏, such that ⋃_𝜎∈∑𝑈_𝜎(𝑡,𝜏)𝐵⊆𝐵₀ for all 𝑡≥𝑡₀. A set 𝑌⊂𝐸 is said to be uniformly attracting for the family of process {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈∑ if for any fixed 𝜏∈ℝ and every 𝐵∈ℬ(𝐸), lim_𝑡→+∞sup_𝜎∈∑dist_𝐸𝑈_𝜎(𝑡,𝜏)𝐵,𝑌=0. (11)

Definition 3. A closed set 𝒜_∑⊂𝐸 is called the uniform attractor of the family of process {𝑈_𝜎(𝑡,𝜏)}, 𝜎∈∑ if it is uniformly attracting (attracting property), and it is contained in any closed uniformly attracting set 𝒜^′ of the family of process {𝑢_𝜎(𝑡,𝜏)}, 𝜎∈∑∶𝒜_𝐸⊆𝒜^′ (minimality property).

Lemma 4. Let ∑ be a compact metric space, and suppose {𝑇(ℎ)∣ℎ≥0} is a family of operators acting on ∑, satisfying the following:
(i) 𝑇(ℎ)=,∀ℎ∈ℝ₊; (12)
(ii) translation identity: 𝑈_𝜎(𝑡+ℎ,𝜏+ℎ)=𝑈_{𝑇(ℎ)𝜎}(𝑡,𝜏),∀𝜎∈,𝑡≥𝜏,𝜏∈ℝ,ℎ≥0, (13) where 𝑈_𝜎(𝑇,𝜏) is arbitrarily a process in compact metric space 𝐸. Moreover, if the family of processes {𝑈_𝜎∈∑(𝑇,𝜏)} is (𝐸×∑,𝐸) continuous, and it has a uniform compact attracting set, then the skew product flow corresponding to it has a global attractor 𝒜 on 𝐸×∑, and the projection of 𝒜 on ∑, 𝒜_∑ is the compact uniform attractor of {𝑈_𝜎∈∑(𝑇,𝜏)}.

Remark 5. Assumption (13) holds if the system has a unique solution.

For brevity, we rewrite system (2)~(5) in the vector form by introducing 𝑊(𝑥,𝑡)=(𝑢(𝑥,𝑡),𝑛(𝑥,𝑡)) and 𝑌(𝑥,𝑡)=(𝑎(𝑥,𝑡),𝑏(𝑥,𝑡)). We denote by 𝐸₀=𝐻²⋂𝐻¹₀(Ω)×𝐻¹₀(Ω) the space of vector functions 𝑊(𝑥,𝑡)=(𝑢(𝑥,𝑡),𝑛(𝑥,𝑡)) with norm ‖𝑊‖_𝐸0=‖𝑢‖²_𝐻2+‖𝑛‖²_𝐻1^1/2. (14) Similarly, we denote by ∑₀ the space of 𝑌(𝑥,𝑡) with norm ‖𝑌‖_∑0=‖𝑎‖²_𝐻1+‖𝑏‖²_𝐻1^1/2. (15) Then system (2)~(5) can be considered as 𝜕_𝑡𝑊=𝐴𝑊+𝜎(𝑡),𝑊∣_𝑡=𝜏=𝑢_𝜏,𝑛_𝜏=𝑊_𝜏,𝑊∣_𝜕Ω=0, (16) where 𝜎(𝑡)=𝑌(𝑥,𝑡) is the symbol of (16).

Assumption 1. Assume that the symbol 𝜎(𝑡) comes from the symbol space ∑ defined by ={𝑌₀(𝑥,𝑠+𝑟)∣𝑟∈ℝ₊}, (17) where 𝑌₀=(𝑎₀(𝑥,𝑡),𝑏₀(𝑥,𝑡))∈𝐿²_𝑐(ℝ;𝐸₀) and the closure is taken in the sense of local quadratic mean convergence topology in the topological space 𝐿²_loc(ℝ;∑₀). Moreover, we suppose that 𝑎_0𝑡(𝑥,𝑡)∈𝐿²_𝑏(ℝ;𝐻¹).

Remark 6. By the conception of translation compact/boundedness we remark that(i)∀𝑌₁∈∑,  ‖𝑌₁‖²_{𝐿2𝑏(ℝ;∑0)}≤‖𝑌₀‖²_{𝐿2𝑏(ℝ;∑0)};(ii)𝑇(𝑡)∑=∑,∀𝑡∈ℝ, where 𝑇(𝑡)𝜑(𝑠)=𝜑(𝑠+𝑡) is an translation operator.

3. Uniform a Priori Estimates in Time

In this section, we derive uniform a priori estimates in time which enable us to show the existence of solutions and the uniform attractor. First we recall the following interpolation inequality (cf. [21]).

Lemma 7. Let 𝑗,𝑚∈ℕ∪{0},𝑞, 𝑟∈ℝ⁺, such that 0≤𝑗<𝑚, 1≤𝑞, 𝑟≤∞. Then one has 𝐷^𝑗𝑢_𝑝≤𝐶𝐷^𝑚𝑢^𝑎_𝑟‖𝑢‖_𝑞^1−𝑎, (18) for 𝑢∈𝑊^𝑚,𝑟(Ω)∩𝐿^𝑞(Ω), where Ω⊂ℝ¹, 𝑗/𝑚≤𝑎≤1, and 1/𝑝=𝑗+𝑎(1/𝑟−𝑚)+1−𝑎/𝑞.

Lemma 8. If 𝑢_𝜏(𝑥)∈𝐿²(Ω) and 𝑌(𝑥,𝑡)∈Σ, then for the solutions of problem (2)~(5), one has ‖𝑢(𝑡)‖≤𝐶₁,∀𝑡≥𝑡₁, (19) where 𝐶₁=𝐶(𝛼,𝑎₀), 𝑡₁=𝐶(𝛼,𝑎₀,‖𝑢_𝜏‖).

Proof. Taking the inner product of (2) with 𝑢 in 𝐻 we get that 𝑖𝑢_𝑡+𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²𝑢,𝑢=(𝑎(𝑥,𝑡),𝑢). (20) Taking the imaginary part of (20), we obtain that 12𝑑‖𝑢‖𝑑𝑡²+𝛼‖𝑢‖²=Im(𝑎(𝑥,𝑡),𝑢). (21) By Young inequality and Remark 6 we have 𝑑‖𝑢‖𝑑𝑡²+𝛼‖𝑢‖²≤1𝛼‖𝑎(𝑥,𝑡)‖²_{𝐿2𝑏(ℝ;𝐻1)}≤1𝛼𝑎₀(𝑥,𝑡)²_{𝐿2𝑏(ℝ;𝐻1)}. (22) And then by Gronwall lemma we can complete the proof.

In the following, we denote that ∫⋅𝑑𝑥=∫_Ω⋅𝑑𝑥, which will not cause confusions.

Lemma 9. Under assumptions of (6), (7) and Assumption 1, if 𝑊(𝜏)∈𝐻¹×𝐻, solutions of problem (2)~(5) satisfy ‖𝑊(𝑡)‖²_𝐻1×𝐻≤𝐶₂,∀𝑡≥𝑡₂, (23) where 𝐶₂=𝐶(𝛼,𝛽,𝑓,𝑔,𝑌₀,𝑎_0𝑡) and 𝑡₂=𝐶(𝛼,𝛽,𝑓,𝑔,𝑌₀,𝑎_0𝑡,‖𝑊_𝜏‖_𝐻¹_×𝐻).

Proof. Taking the inner product of (2) with 𝑢_𝑡 in 𝐻 and taking the real part, we get that −12𝑑𝑢𝑑𝑡^𝑥2−12𝑛𝑑|𝑢|𝑑𝑡²𝑑𝑥+Re𝑖𝛼𝑢,𝑢_𝑡+12𝑔|𝑢|²𝑑|𝑢|𝑑𝑡²𝑑𝑥=Re𝑎(𝑥,𝑡),𝑢_𝑡. (24) By (3) we know that 𝑑𝑛𝑑𝑑𝑡|𝑢|𝑑𝑡²=𝑑𝑑𝑥𝑛|𝑢|𝑑𝑡²𝑑𝑥−|𝑢|²𝑛_𝑡=𝑑𝑑𝑥𝑛|𝑢|𝑑𝑡²𝑑𝑥+|𝑢|²|𝑢|²_𝑥𝑑𝑥+𝛽𝑛|𝑢|²+𝑓|𝑢|𝑑𝑥₂|𝑢|²𝑑𝑥−𝑏(𝑥,𝑡)|𝑢|²=𝑑𝑑𝑥𝑛|𝑢|𝑑𝑡²𝑑𝑥+𝛽𝑛|𝑢|²+𝑓|𝑢|𝑑𝑥₂|𝑢|²𝑑𝑥−𝑏(𝑥,𝑡)|𝑢|²𝑑𝑥, (25) which shows that −12𝑑𝑢𝑑𝑡^𝑥2−12𝑑𝑛|𝑢|𝑑𝑡²𝑑𝑥+𝛽𝑛|𝑢|²+𝑓|𝑢|𝑑𝑥₂|𝑢|²𝑑𝑥−𝑏(𝑥,𝑡)|𝑢|²+1𝑑𝑥2𝑑𝐺|𝑢|𝑑𝑡²𝑑𝑥+Re𝑖𝛼𝑢,𝑢_𝑡−𝑑(𝑎())𝑑𝑡Re𝑥,𝑡,𝑢+Re𝑎_𝑡(𝑥,𝑡)𝑢𝑑𝑥=0, (26) where 𝐺(𝑠) is introduced by 𝐺(𝑠)=^𝑠₀𝑔(𝜉)𝑑𝜉. (27) Taking the inner product of (2) with 𝑢 in 𝐻 and taking the real part, we get that Re𝑖𝑢_𝑡,𝑢−𝑢_𝑥²−𝑛|𝑢|²|𝑢|𝑑𝑥+𝑔²|𝑢|²𝑑𝑥−Re(𝑎(𝑥,𝑡),𝑢)=0. (28) Multiply (28) by 𝛼, and add the resulting identity to (26) to get −12𝑑𝑢𝑑𝑡^𝑥2−12𝑑𝑛|𝑢|𝑑𝑡²−1𝑑𝑥2𝛽+2𝛼𝑛|𝑢|²1𝑑𝑥−2𝑓|𝑢|²|𝑢|²+1𝑑𝑥2𝑏(𝑥,𝑡)|𝑢|²1𝑑𝑥+2𝑑𝐺|𝑢|𝑑𝑡²𝑑𝑥−𝛼𝑢_𝑥²+𝛼𝑔|𝑢|²|𝑢|²(𝑎())−𝑑𝑑𝑥−𝛼Re𝑥,𝑡,𝑢(𝑎())𝑎𝑑𝑡Re𝑥,𝑡,𝑢+Re_𝑡(𝑥,𝑡)𝑢𝑑𝑥=0. (29) That is, 𝑑𝑢𝑑𝑡^𝑥2+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑑𝑥+2Re𝑎(𝑥,𝑡)𝑢𝑢𝑑𝑥+𝛼_𝑥²+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑎()𝑑𝑥+2Re𝑥,𝑡𝑢𝑑𝑥+𝛼𝑢_𝑥²=−𝑓|𝑢|²|𝑢|²𝑑𝑥+𝑏(𝑥,𝑡)|𝑢|²𝐺|𝑢|𝑑𝑥−𝛼²𝑑𝑥−𝛼+𝛽𝑛|𝑢|²𝑎𝑑𝑥+2Re_𝑡(𝑥,𝑡)𝑢𝑑𝑥. (30) In the following, we denote by 𝐶 any constants depending only on the data (𝛼,𝛽,𝑓,𝑔), and 𝐶(⋅,⋅) means it depends not only on (𝛼,𝛽,𝑓,𝑔) but also on parameters in the brackets. ∀𝜌>0, when 𝑡 is sufficiently large, by (6) and Lemmas 7 and 8, we have −𝑓|𝑢|²|𝑢|²|𝑢|𝑑𝑥≤𝐶^𝑝+2𝑑𝑥+𝐶|𝑢|²|𝑢|𝑑𝑥≤𝐶^2𝑝+|𝑢|⁴𝑑𝑥+𝐶‖𝑢‖²≤𝐶‖𝑢‖⁴₄+𝐶≤𝐶𝑢_𝑥‖𝑢‖³𝑢+𝐶≤𝜌_𝑥²𝜌,𝑏(𝑢|+𝐶𝑥,𝑡)|²≤‖‖𝑑𝑥𝑏(𝑥,𝑡)²_{𝐿2𝑏(ℝ;∑0)}+‖𝑢‖⁴₄≤𝑌₀²_{𝐿2𝑏(ℝ;∑0)}+𝐶𝑢_𝑥≤𝜌𝑢_𝑥²+𝐶₂𝜌. (31) By (6) we deduce that |𝐺(𝑠≤2)|3𝑐₂𝑠^3/2+𝑐₂𝑠,∀𝑠≥0. (32) And then −𝛼_Ω𝐺|𝑢|²|𝑢|𝑑𝑥≤𝐶³+|𝑢|²𝑑𝑥≤𝐶‖𝑢‖³₃+𝐶‖𝑢‖²≤𝐶𝑢_𝑥^1/2‖𝑢‖^5/2+𝐶≤𝜌𝑢_𝑥²+𝐶₃𝜌,−𝛼+𝛽_Ω𝑛|𝑢|²‖𝑛‖𝑑𝑥≤𝜌²+𝐶𝜌‖𝑢‖⁴₄≤𝜌‖𝑛‖²+𝜌𝑢_𝑥²+𝐶₄𝜌,2Re_Ω𝑎_𝑡(𝑥,𝑡)≤𝑎𝑢𝑑𝑥_𝑡(𝑥,𝑡)²_{𝐿2𝑐(ℝ;Σ0)}+‖𝑢‖²≤𝐶𝑎_0𝑡²_{𝐿2𝑐(ℝ;Σ0)},‖𝑢‖². (33)(34)(35) By (30)~(35) we get that 𝑑𝑢𝑑𝑡^𝑥2+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑑𝑥+2Re𝑎(𝑥,𝑡)𝑢𝑢𝑑𝑥+𝛼_𝑥²+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑎()𝑑𝑥+2Re𝑥,𝑡𝑢𝑑𝑥+𝛼𝑢_𝑥²≤𝜌‖𝑛‖²+4𝜌𝑢_𝑥²+𝐶𝜌+𝐶𝑎_0𝑡²_{𝐿2𝑏(ℝ;Σ0)},‖𝑢‖². (36) Similarly we can also deduce that 𝑑𝑢𝑑𝑡^𝑥2+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑑𝑥+2Re𝑎(𝑥,𝑡)𝑢𝑢𝑑𝑥+𝛽_𝑥²+𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑎()𝑑𝑥+2Re𝑥,𝑡𝑢𝑑𝑥+2𝛼−𝛽𝑢_𝑥²≤𝜌‖𝑛‖²+4𝜌𝑢_𝑥²+𝐶𝜌+𝐶𝑎_0𝑡²_{𝐿2𝑏(ℝ;Σ0)},‖𝑢‖². (37) Taking the inner product of (3) with 𝑛 in 𝐻, we see that 12𝑑‖𝑛‖𝑑𝑡²+𝑛|𝑢|²_𝑥𝑑𝑥+𝛽‖𝑛‖²+𝑓|𝑢|²𝑛𝑑𝑥−𝑏(𝑥,𝑡)𝑛𝑑𝑥=0. (38) By (2) we get that 𝑛|𝑢|²_𝑥𝑑𝑥=𝑛𝑢_𝑥𝑢𝑑𝑥+𝑛𝑢𝑢_𝑥𝑢𝑑𝑥=𝑖_𝑡𝑢_𝑥−𝑢_𝑡𝑢_𝑥𝑑𝑥+2Re𝑖𝛼𝑢𝑢_𝑥𝑏()𝑑𝑥−2Re𝑥,𝑡𝑢_𝑥𝑑𝑑𝑥,𝑑𝑡𝑖𝑢𝑢_𝑥−𝑖𝑢_𝑥𝑢𝑢𝑑𝑥=𝑖_𝑡𝑢_𝑥+𝑢𝑢_𝑥𝑡−𝑢_𝑥𝑡𝑢−𝑢_𝑥𝑢_𝑡𝑢𝑑𝑥=𝑖_𝑡𝑢_𝑥−𝑢_𝑥𝑢_𝑡+𝑢_𝑡𝑢_𝑥−𝑢_𝑥𝑢_𝑡𝑢𝑑𝑥=2𝑖_𝑡𝑢_𝑥−𝑢_𝑡𝑢_𝑥𝑑𝑥. (39)(40) It comes from (38)~(40) that 𝑑‖𝑛‖𝑑𝑡²+𝑑𝑖𝑢𝑑𝑡𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥+2𝛽²+𝑖𝛼𝑢𝑢_𝑥−𝑢_𝑥𝑢𝑢𝑑𝑥≤𝑖𝛼𝑢_𝑥−𝑢_𝑥𝑢𝑑𝑥−4Re𝑖𝛼𝑢𝑢_𝑥𝑎()𝑑𝑥+4Re𝑥,𝑡𝑢_𝑥𝑓|𝑢|𝑑𝑥−2²𝑛𝑑𝑥+2𝑏(𝑥,𝑡)𝑛𝑑𝑥. (41) Deal with the right hand side of inequality (41), by Lemmas 7 and 8, 𝑖𝛼𝑢𝑢_𝑥−𝑢_𝑥𝑢𝑑𝑥≤2𝛼‖𝑢‖𝑢_𝑥≤𝜌𝑢_𝑥+𝐶₁𝜌,−4Re𝑖𝛼𝑢𝑢_𝑥𝑑𝑥≤4𝛼‖𝑢‖𝑢_𝑥≤𝜌𝑢_𝑥+𝐶₂𝜌𝑓|𝑢|,−2²|𝑢|𝑛𝑑𝑥≤𝐶^𝑝|𝑛||𝑛|≤1𝑑𝑥+𝐶𝑑𝑥2𝜌‖𝑛‖²+𝐶𝜌|𝑢|^2𝑝1𝑑𝑥+2𝜌‖𝑛‖²𝜌‖𝑛‖+𝐶≤𝜌²+𝐶𝜌𝑢_𝑥^𝑝−1‖𝑢‖^𝑝+1𝜌‖𝑛‖+𝐶≤𝜌²+𝜌𝑢_𝑥²+𝐶₃𝜌,4Re𝑎(𝑥,𝑡)𝑢_𝑥‖𝑎(𝑑𝑥≤4𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)}𝑢_𝑥≤𝜌𝑢_𝑥+𝐶₄𝜌,𝑎₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)},()‖𝑏(2𝑏𝑥,𝑡𝑛𝑑𝑥≤2𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)}‖𝑛‖≤𝜌‖𝑛‖²+𝐶₅𝜌,𝑏₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)}. (42)(43)(44)(45)(46) So 𝑑‖𝑛‖𝑑𝑡²+𝑖𝑢𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥+2𝛽²+𝑖𝛼𝑢𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥≤2𝜌²+4𝜌𝑢_𝑥²+𝐶𝜌,𝑌₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)}. (47) Analogously, we can also deduce that 𝑑‖𝑛‖𝑑𝑡²+𝑖𝑢𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥+2𝛽²+𝑖𝛽𝑢𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥≤2𝜌²+4𝜌𝑢_𝑥²+𝐶𝜌,𝑌₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)}. (48) Set 𝛾=min{𝛼,𝛽}, and 𝐸=𝑢_𝑥²+‖𝑛‖²+𝑛|𝑢|²|𝑢|𝑑𝑥−𝐺²𝑑𝑥+2Re𝑎𝑢𝑢𝑑𝑥+𝑖𝑢_𝑥−𝑢_𝑥𝑢𝑑𝑥. (49) Then by (36), (47) and (37), (48) we can, respectively, get 𝑑𝑢𝑑𝑡𝐸+𝛼𝐸+𝛼_𝑥²+𝛽‖𝑛‖²≤8𝜌𝑢_𝑥²+3𝜌‖𝑛‖²+𝐶𝜌,Y₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)},𝑑𝑢𝑑𝑡𝐸+𝛽𝐸+𝛼_𝑥²+𝛽‖𝑛‖²≤8𝜌𝑢_𝑥²+3𝜌‖𝑛‖²+𝐶𝜌,𝑌₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)}, (50) which shows that if we set 𝜌≤min{𝛼/8,𝛽/3}, we can deduce that 𝑑𝑑𝑡𝐸+𝛾𝐸≤𝐶₀,∀𝑡≥𝑡₀, (51) where 𝐶₀=𝐶(𝜌,‖𝑌₀(𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)},‖𝑎_0𝑡(𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)}). By Gronwall lemma we see that 𝐸(𝑡)≤𝐸𝑡₀𝑒^{−𝛾(𝑡−𝑡0)}+𝐶₀𝛾,∀𝑡≥𝑡₀. (52) Similar to (33), (34), (45), and (42), for 𝑡≥𝑡₀ we have 𝑛|𝑢|²𝑑𝑥−𝐺|𝑢|²𝑑𝑥+2Re𝑎𝑢𝑑𝑥+𝑖𝑢𝑢_𝑥−𝑢_𝑥𝑢‖𝑛‖𝑑𝑥≤𝜌²+𝜌𝑢_𝑥²+𝐶𝜌,𝑎₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)}. (53) And then 𝐸𝑡₀≤𝑢_𝑥(𝑡₀)²+𝑛(𝑡₀)²+𝑛𝑡₀𝑢𝑡₀²𝑑𝑥−𝐺𝑢𝑡₀²𝑡𝑑𝑥+2Re𝑎₀𝑢𝑡₀𝑢𝑡𝑑𝑥+𝑖₀𝑢_𝑥𝑡₀−𝑢_𝑥𝑡₀𝑢𝑡₀(𝑅),𝑑𝑥≤𝐶 (54) where 𝐶(𝑅)=𝐶(𝜌,‖𝑌₀(𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)},‖𝑎_0𝑡(𝑥,𝑡)‖_𝐿²_{𝑏(ℝ;Σ0)},𝑅) when ‖𝑊_𝜏‖_𝐻¹_×𝐻≤𝑅. Then by (52) we infer that 𝐸(𝑡)≤𝐶(𝑅)𝑒^{−𝛾(𝑡−𝑡0)}+𝐶₀𝛾,∀𝑡≥𝑡₀,≤2𝐶₀𝛾,∀𝑡≥𝑡_∗, (55) where 𝑡_∗=inf{𝑡∣𝑡≥𝑡₀ and 𝐶(𝑅)𝑒^{−𝛾(𝑡∗−𝑡0)}≤𝐶₀/𝛾}. By (49), (53), and (55) we infer that 𝑢_𝑥(𝑡)²+‖𝑛(𝑡)‖²≤𝜌‖𝑛‖²+𝜌𝑢_𝑥²+𝐶₀. (56) Choose 𝜌=min{𝛼/8,𝛽/3,1/2}; then we have 𝑢_𝑥²+‖𝑛(𝑡)‖²≤𝐶𝑌₀(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)},𝑎_0𝑡(𝑥,𝑡)_{𝐿2𝑏(ℝ;Σ0)},∀𝑡≥𝑡_∗, (57) which concludes the proof by using Lemma 8.

Lemma 10. Under assumptions of Lemma 9, if 𝑊(𝜏)∈𝐸₀=𝐻²×𝐻¹, solutions of problem (2)~(5) satisfy ‖𝑊(𝑡)‖²_𝐸0≤𝐶₂,∀𝑡≥𝑡₃, (58) where 𝐶₂=𝐶(𝛼,𝛽,𝑓,𝑔,𝑌₀,𝑎_0𝑡) and 𝑡₃=𝐶(𝛼,𝛽,𝑓,𝑔,𝑌₀,𝑎_0𝑡,‖𝑊_𝜏‖_𝐸0).

Proof. Taking the real part of the inner product of (2) with 𝑢_𝑥𝑥𝑡 in 𝐻, we have 𝑑𝑢𝑑𝑡^𝑥𝑥2−Re𝑛𝑢𝑢_𝑥𝑥𝑡𝑑𝑥+Re𝑖𝛼𝑢,𝑢_𝑥𝑥𝑡+Re𝑔|𝑢|²𝑢𝑢_𝑥𝑥𝑡𝑎()𝑑𝑥−Re𝑥,𝑡𝑢_𝑥𝑥𝑡𝑑𝑥=0. (59) By (2) and (3), we have −Re𝑛𝑢𝑢_𝑥𝑥𝑡𝑑𝑑𝑥=−𝑑𝑡Re𝑛𝑢𝑢_𝑥𝑥𝑑𝑥+Re𝑛_𝑡𝑢𝑢_𝑥𝑥𝑑𝑥+Re𝑛𝑢_𝑡𝑢_𝑥𝑥𝑑𝑑𝑥=−𝑑𝑡Re𝑛𝑢𝑢_𝑥𝑥𝑢𝑑𝑥−Re𝑢_𝑥𝑥|𝑢|²_𝑥+𝛽𝑛+𝑓|𝑢|²𝑛−𝑏𝑑𝑥+Re𝑢_𝑥𝑥−𝑖𝑛𝑢−𝛼𝑢+𝑖𝑔|𝑢|²𝑢−𝑖𝑎𝑑𝑥. (60) Since Re𝑖𝛼𝑢,𝑢_𝑥𝑥𝑡=Re𝑖𝛼𝑢𝑢_𝑡𝑥𝑥𝑑𝑥=−Re𝑖𝛼𝑢_𝑡𝑢_𝑥𝑥𝑑𝑥, (61) we see that Re𝑖𝛼𝑢,𝑢_𝑥𝑥𝑡=𝛼𝑢_𝑥𝑥²−𝛼Re𝑛𝑢𝑢_𝑥𝑥|𝑢|𝑑𝑥+𝛼Re𝑔²𝑢𝑢_𝑥𝑥𝑎𝑑𝑥−𝛼Re𝑢_𝑥𝑥𝑑𝑥. (62) Multiplying (2) by 𝑢 and taking the real part, we find that |𝑢|²_𝑡=2Re𝑖𝑢_𝑥𝑥𝑢−2𝛼|𝑢|²−2Re(𝑖𝑎𝑢), (63) therefore, Re𝑔|𝑢|²𝑢𝑢_𝑥𝑥𝑡𝑑𝑥=−𝑔^′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡−𝑔|𝑢|𝑑𝑥₂12𝑑𝑢𝑑𝑡^𝑥2𝑔𝑑𝑥=−^′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡−1𝑑𝑥2𝑑𝑔|𝑢|𝑑𝑡²𝑢_𝑥²+𝑔𝑑𝑥_′|𝑢|²𝑢_𝑥²Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢))𝑑𝑥. (64) Now we deal with (64) to get (70). Due to equalities |𝑢|²_𝑥=2Re𝑢𝑢_𝑥,𝑑𝑢𝑑𝑡Re𝑢_𝑥=Re𝑢_𝑡𝑢_𝑥+Re𝑢𝑢_𝑥𝑡, (65) we deduce that 𝑔^′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡=𝑑𝑑𝑥𝑔𝑑𝑡^′|𝑢|²2Re𝑢𝑢_𝑥Re𝑢𝑢_𝑥−𝑔𝑑𝑥_′′|𝑢|²|𝑢|²_𝑡2Re𝑢𝑢_𝑥Re𝑢𝑢_x−𝑔𝑑𝑥_′|𝑢|²2Re𝑢𝑢_𝑥𝑡Re𝑢𝑢_𝑥−𝑔𝑑𝑥_′|𝑢|²2Re𝑢𝑢_𝑥Re𝑢_𝑡𝑢_𝑥𝑑𝑥. (66) We take care of terms in (66) as follows: 𝑔^′′|𝑢|²|𝑢|²_𝑡2Re𝑢𝑢_𝑥Re𝑢𝑢_𝑥𝑑𝑥=4𝑔^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑔𝑑𝑥,_′|𝑢|²2Re𝑢𝑢_𝑥𝑡Re𝑢𝑢_𝑥𝑑𝑥=𝑔^′|𝑢|²2Re𝑢_𝑡𝑢_𝑥Re𝑢𝑢_𝑥+𝑔𝑑𝑥_′|𝑢|²2Re𝑢𝑢_𝑥𝑡Re𝑢𝑢_𝑥𝑑𝑥=2𝑔^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²+𝑔𝑢−𝑎𝑑𝑥_′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡𝑔𝑑𝑥,_′|𝑢|²2Re𝑢𝑢_𝑥Re𝑢_𝑡𝑢_𝑥𝑔𝑑𝑥=2^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²𝑢−𝑎𝑑𝑥. (67) It follows from (66)~(67) that −𝑔^′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡𝑑𝑑𝑥=−𝑔𝑑𝑡^′|𝑢|²2Re𝑢𝑢_𝑥Re𝑢𝑢_𝑥𝑔𝑑𝑥+4^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑔𝑑𝑥+4^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²+𝑔𝑢−𝑎𝑑𝑥_′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡𝑑𝑥. (68) And then −𝑔^′|𝑢|²|𝑢|²_𝑥Re𝑢𝑢_𝑥𝑡𝑑𝑑𝑥=−𝑔𝑑𝑡^′|𝑢|²Re𝑢𝑢_𝑥Re𝑢𝑢_𝑥𝑔𝑑𝑥+2^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑑𝑥+2𝑔^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²𝑢−𝑎𝑑𝑥. (69) From (64) and (69) we have Re𝑔|𝑢|²𝑢𝑢_𝑥𝑥𝑡𝑑𝑑𝑥=−𝑔𝑑𝑡^′|𝑢|²Re𝑢𝑢_𝑥Re𝑢𝑢_𝑥−1𝑑𝑥2𝑑𝑔|𝑢|𝑑𝑡²𝑢_𝑥²𝑔𝑑𝑥+2^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑑𝑥+2𝑔^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²+𝑔𝑢−𝑎𝑑𝑥_′|𝑢|²𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑑𝑥. (70) By (59), (60), (62), and (70) we conclude that 𝑑𝑢𝑑𝑡^𝑥𝑥2−2Re𝑛𝑢𝑢_𝑥𝑥−2𝑔^′|𝑢|²Re𝑢𝑢_𝑥²−𝑔|𝑢|²|𝑢|²_𝑥−2Re𝑎𝑢_𝑥𝑥+2𝛼𝑢_𝑥𝑥²−2Re𝑛𝑢𝑢_𝑥𝑥−2𝑔^′|𝑢|²Re𝑢𝑢_𝑥²−𝑔|𝑢|²𝑢_𝑥²−2Re𝑎𝑢_𝑥𝑥+2𝛼𝑛𝑢𝑢_𝑥𝑥𝑑𝑥+4𝛼𝑔^′|𝑢|²Re𝑢𝑢_𝑥²𝑔|𝑢|𝑑𝑥+2𝛼²𝑢_𝑥²𝑑𝑥+2𝛼Re𝑎𝑢_𝑥𝑥𝑢𝑑𝑥−2Re𝑢_𝑥𝑥|𝑢|²_𝑥+𝛽𝑛+𝑓|𝑢|²𝑛−𝑏𝑑𝑥+2Re𝑢_𝑥𝑥−𝑖𝑛𝑢−𝛼𝑢+𝑖𝑔|𝑢|²|𝑢|−𝑖𝑎𝑑𝑥+2𝛼Re𝑔²𝑢𝑢_𝑥𝑥𝑔𝑑𝑥+4^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑑𝑥+4𝑔^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥−𝑛𝑢+𝑖𝛼𝑢+𝑔|𝑢|²𝑔𝑢−𝑎𝑑𝑥+2^′|𝑢|²𝑢_𝑥²Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑑𝑥+2Re𝑎_𝑡𝑢_𝑥𝑥𝑑𝑥=0, (71) where ∫⋅=∫⋅𝑑𝑥.
For later purpose, we let 𝐹(𝑢,𝑛)=−2Re𝑛𝑢𝑢_𝑥𝑥𝑑𝑥−2𝑔^′|𝑢|²Re𝑢𝑢_𝑥²𝑑𝑥−𝑔|𝑢|²𝑢_𝑥²𝑎𝑑𝑥−2Re𝑢_𝑥𝑥()𝑑𝑥,−𝐺𝑢,𝑛=2𝛼𝑛𝑢𝑢_𝑥𝑥𝑑𝑥+4𝛼𝑔^′|𝑢|²Re𝑢𝑢_𝑥²𝑔|𝑢|𝑑𝑥+2𝛼²𝑢_𝑥²𝑑𝑥+2𝛼Re𝑎𝑢_𝑥𝑥𝑑𝑥−2Re𝑢𝑢_𝑥𝑥|𝑢|²_𝑥+𝛽𝑛+𝑓|𝑢|²𝑛−𝑏𝑑𝑥+2Re𝑢_𝑥𝑥−𝑖𝑛𝑢−𝛼𝑢+𝑖𝑔|𝑢|²𝑢|−𝑖𝑎𝑑𝑥+2𝛼Re𝑔|²𝑢𝑢_𝑥𝑥𝑔𝑑𝑥+4^′′|𝑢|²Re𝑢𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑔𝑑𝑥+4^′|𝑢|²Re𝑢𝑢_𝑥×Re𝑖𝑢_𝑥𝑢_𝑥𝑥|𝑢|−𝑛𝑢+𝑖𝛼𝑢+𝑔²𝑔𝑢−𝑎𝑑𝑥+2^′|𝑢|²𝑢_𝑥²×Re𝑖𝑢_𝑥𝑥𝑢−𝛼|𝑢|²−Re(𝑖𝑎𝑢)𝑎𝑑𝑥+2Re_𝑡𝑢_𝑥𝑥𝑑𝑥. (72)(73) Then from (71) we have 𝑑𝑢𝑑𝑡^𝑥𝑥2+𝐹+2𝛼𝑢_𝑥𝑥²+𝐹=𝐺, (74) or 𝑑𝑢𝑑𝑡^𝑥𝑥2+𝐹+𝛼𝑢_𝑥𝑥²+𝐹+𝛼𝑢_𝑥𝑥²=𝐺−𝛼𝐹. (75) By Lemma 9 and Agmon inequality we have ‖𝑢(𝑡)‖²_𝐻1+‖𝑢(𝑡)‖²_∞+‖𝑛(𝑡)‖²_𝐻≤2𝐶₂,∀𝑡≥𝑡₂. (76) In the following, we denote by 𝐶=𝐶(𝛼,𝛽,𝑓,𝑔,𝑌₀,𝑎_0𝑡). By Lemma 7 and (76) we estimate the size of |𝐺−𝛼𝐹| to get 𝑑𝑢𝑑𝑡^𝑥𝑥2+𝐹+𝛼𝑢_𝑥𝑥²+𝐹+𝛼𝑢_𝑥𝑥²≤𝐶|𝑛|²𝑢_𝑥𝑥𝑑𝑥+𝐶𝑢_𝑥²𝑢_𝑥𝑥𝑢𝑑𝑥+𝐶_𝑥²||𝑢𝑛𝑢𝑑𝑥+𝐶≤𝐶_𝑥𝑥‖𝑛‖²₄+𝐶𝑢_𝑥𝑥𝑢_𝑥²₄+𝐶‖𝑢‖_𝐿^∞‖𝑛‖𝑢_𝑥²₄𝑢+𝐶≤𝐶_𝑥𝑥𝑛_𝑥^1/2‖𝑛‖^3/2+𝐶𝑢_𝑥𝑥^3/2𝑢_𝑥^3/2+𝐶‖𝑢‖_𝐿^∞‖𝑛‖𝑢_𝑥𝑥^7/4‖𝑢‖^1/4𝑢+𝐶≤𝐶_𝑥𝑥𝑛_𝑥^1/2+𝐶𝑢_𝑥𝑥^7/4≤𝛼+𝐶2𝑢_𝑥𝑥²+𝛽2𝑛_𝑥²+𝐶. (77) Taking the inner product of (3) with 𝑛_𝑥𝑥 in 𝐻, we see that −12𝑑𝑛𝑑𝑡^𝑥2+|𝑢|²_𝑥𝑛_𝑥𝑥𝑑𝑥−𝛽𝑛_𝑥²+𝑓|𝑢|²𝑛_𝑥𝑥𝑑𝑥−𝑏𝑛_𝑥𝑥𝑑𝑥=0. (78) Since |𝑢|²_𝑥𝑛_𝑥𝑥𝑑𝑥=2Re𝑢𝑢_𝑥𝑛_𝑥𝑥𝑢𝑑𝑥=−2Re𝑢_𝑥𝑥𝑛_𝑥+𝑢_𝑥²𝑛_𝑥𝑑𝑥, (79) by (78) we can deduce that 𝑑𝑛𝑑𝑡^𝑥2+4Re𝑢𝑢_𝑥𝑥𝑛_𝑥𝑢𝑑𝑥+4_𝑥²𝑛_𝑥𝑑𝑥+2𝛽𝑛_𝑥²+2𝑓^′|𝑢|²𝑢_𝑥𝑢+𝑢𝑢_𝑥𝑛_𝑥𝑏𝑑𝑥−2_𝑥𝑛_𝑥𝑑𝑥=0. (80) From (2) we know that 𝑖𝑢_𝑡𝑥+𝑢_𝑥𝑥𝑥−𝑛_𝑥𝑢−𝑛𝑢_𝑥+𝑖𝛼𝑢_𝑥+𝑔^′|𝑢|²|𝑢|²_𝑥𝑢+𝑔|𝑢|²𝑢_𝑥−𝑎_𝑥(𝑥,𝑡)=0. (81) Taking the real part of the inner product to (81) with 𝑢_𝑥𝑥 in 𝐻, we have Re𝑖𝑢_𝑡𝑥𝑢_𝑥𝑥−Re𝑛_𝑥𝑢𝑢_𝑥𝑥𝑑𝑥−Re𝑛𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+Re𝑖𝛼𝑢_𝑥𝑢_𝑥𝑥𝑔𝑑𝑥+Re^′|𝑢|²|𝑢|²_𝑥𝑢𝑢_𝑥𝑥|𝑢|𝑑𝑥+Re𝑔²𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥−Re𝑎_𝑥𝑢_𝑥𝑥𝑑𝑥=0. (82) Because of 𝑑𝑑𝑡Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥=2Re𝑖𝑢_𝑡𝑥𝑢_𝑥𝑥𝑑𝑥, (83) it holds that 12𝑑𝑑𝑡Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥−Re𝑛_𝑥𝑢𝑢_𝑥𝑥𝑑𝑥−Re𝑛𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+Re𝑖𝛼𝑢_𝑥𝑢_𝑥𝑥𝑔𝑑𝑥+Re^′|𝑢|²|𝑢|²_𝑥𝑢𝑢_𝑥𝑥𝑔|𝑢|𝑑𝑥+Re²𝑢_𝑥𝑢_𝑥𝑥𝑎𝑑𝑥−Re_𝑥𝑢_𝑥𝑥𝑑𝑥=0. (84) By (84) and (80), we find that 𝑑𝑛𝑑𝑡^𝑥2𝑑+2𝑑𝑡Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+4𝑢_𝑥²𝑛_𝑥𝑑𝑥+2𝛽𝑛_𝑥²+2𝑓^′|𝑢|²|𝑢|²_𝑥𝑛_𝑥𝑏𝑑𝑥−2_𝑥𝑛_𝑥𝑑𝑥−4Re𝑛𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+4𝛼Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+4Re𝑔^′|𝑢|²|𝑢|²_𝑥𝑢𝑢_𝑥𝑥𝑔|𝑢|𝑑𝑥+4Re²𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥−4Re𝑎_𝑥𝑢_𝑥𝑥𝑑𝑥=0. (85) That is, 𝑑𝑛𝑑𝑡^𝑥2+2Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑛𝑑𝑥+2𝛽_𝑥²+2Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥=4𝛽Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥−4𝑢_𝑥²𝑛_𝑥𝑓𝑑𝑥−2^′|𝑢|²|𝑢|²_𝑥𝑛_𝑥𝑑𝑥+2𝑏_𝑥𝑛_𝑥𝑑𝑥+4Re𝑛𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥−4𝛼Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑔𝑑𝑥−4Re^′|𝑢|²|𝑢|²_𝑥𝑢𝑢_𝑥𝑥𝑔|𝑢|𝑑𝑥−4Re²𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+4Re𝑎_𝑥𝑢_𝑥𝑥𝑑𝑥. (86) Similar to (77), we estimate each term in (86), and then we get 𝑑𝑛𝑑𝑡^𝑥2+2Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑛𝑑𝑥+𝛽_𝑥²+2Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+𝛽𝑛_𝑥²≤2𝛽Re𝑖𝑢_𝑥𝑢_𝑥𝑥𝑑𝑥+𝐶𝑢_𝑥𝑥+𝐶𝑢_𝑥²₄𝑛_𝑥+𝐶𝑛_𝑥+𝐶𝑢_𝑥∞‖𝑛‖𝑢_𝑥𝑥≤𝐶𝑢_𝑥𝑥+𝐶𝑢_𝑥^3/2𝑢_𝑥𝑥^1/2𝑛_𝑥+𝐶𝑛_𝑥+𝐶‖𝑢‖^1/4𝑢_𝑥𝑥^3/4‖𝑛‖𝑢_𝑥𝑥≤𝛼2𝑢_𝑥𝑥²+𝛽