# Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations

Authors: Hongyong Cui; Jie Xin
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.


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## Body

## 1. Introduction

In this paper, we investigate the long time behavior of solutions for the following nonautonomous generalized dissipative LS equations with quasiperiodic forces:(1)iut+uxx-nu+iγu+g(|u|2)u=h1(x,t),(2)nt+βn+|u|x2+f(|u|2)=h2(x,t),
with initial conditions and space-periodic boundary conditions as
(3)u(x,τ)=uτ(x),n(x,τ)=nτ(x),(4)u(x-D,t)=u(x+D,t),n(x-D,t)=(x+D,t),
where x∈Ω=(-D,D),D>0 and γ and β are positive constants.The long-wave short-wave (LS) resonance equations arise in many kinds of physical models (see [1–4]). Due to their rich physical and mathematical properties, the LS equations have drawn much attention. The autonomous situations, including the existence of solutions, the solitary waves and their stability, and the long time behaviors of the solutions, have been deeply researched (see [5–14]).Recently, the nonautonomous case of LS equations with translation compact forces was studied in [15]. Because of the nonlinear resonance of the equations, it is difficult to prove the continuity of the process U(t,τ) generated by (1)–(4). Thus, it is hard to construct the uniform attractor directly by constructing a compact uniform absorbing set even if the forces are translation compact, and in [15] only the weakly compact uniform attractor for the system is obtained.In this paper, we firstly investigate the compact uniform attractor for systems (1)–(4) by employing the energy equations and the energy method presented by Ball (see [16, 17]). The energy method can be concisely understood as the following two steps (e.g., in autonomous cases): (1) construct a weakly compact attractor and (2) prove the strong compactness of the weak attractor, that is, verify that the weak attractor is actually the strong one. To accomplish Step 1, one can construct a bounded (weakly compact) absorbing set and the weak continuity of the system. Step 2 is usually deduced by applying proper energy inequalities together with Lemmas 11 and 12. Obviously, this method is good at solving problems which are not continuous and those that lack Sobolev embeddings (such as systems defined in unbounded domains).Besides, approximate inertial manifolds (AIM) for the system is studied afterwards. This manifold is a finite-dimensional smooth surface in a phase space, whose small vicinity attracts all the trajectories at a much higher speed than global attractors. To investigate AIM, by employing the extending phase space method we transfer the nonautonomous systemUσ(t·τ) to an autonomous one S(t), and we get the AIM for Uσ(t·τ) by constructing the AIM for S(t).Themain result of this paper contains Theorems 13 and 17. It is summarized by the following.Main Theorem.  Assume that(i)
hj(x,t),j=1,2,
are quasiperiodic forces satisfying Assumption1;(ii)
generalizedf(s)andg(s)are quasilinear functions satisfying (9) and (10).Then systems (1)–(4) generate a family of processesUσ(t,τ)inE0:=Hper2(Ω)×Hper1(Ω).Moreover, the family of processesUσ(t,τ)admits a compact uniform attractor𝒜Σand an AIM inE0.We would like to remark that the existence of the compact uniform attractor for the system does not depend heavily on the quasiperiodicity of the forces. It still holds when the forces are just translation bounded (see Remark14), that is, it strengthens the result in [15].This paper is organized as follows. In Section2, we show the LS equations in details and we deeply introduce the quasiperiodicity conditions. In Section 3, we get the uniform a prior estimates for the solutions. In Section 4, we study the unique existence of the solution. In Section 5 the existence of the uniform attractor for (5)–(8) is obtained by applying weak convergence method. In Section 6, AIM for (5)–(8) is constructed by extending and splitting the phase space and making use of projection operators.Throughout this paper, we denote by∥·∥ the norm of H=Lper2(Ω) with usual inner product (·,·), denote by ∥·∥p the norm of Lperp(Ω) for all 1⩽p⩽∞(∥·∥2=∥·∥), and denote by ∥·∥X the norm of any Banach space X. Besides, notations “⇀”, “⇀*”, and “→” denote weakly converges to, weakly star converges to and strongly converges to, respectively. And we denote different constants by the same letter C, and C(·,·) represents that the constant relies only on the parameters that appear in the brackets.


## 2. Preliminaries

We show the nonautonomous dissipative generalized long-short wave equations with quasiperiodic forces in details as follows:(5)iut+uxx-nu+iγu+g(|u|2)u=h1(x,t),(6)nt+βn+|u|x2+f(|u|2)=h2(x,t),
with initial value conditions
(7)u(x,τ)=uτ(x),n(x,τ)=nτ(x),x∈Ω=(-D,D),D>0,
and periodic boundary value conditions
(8)u(x-D,t)=u(x+D,t),n(x-D,t)=(x+D,t),
where u(x,t) is an unknown complex valued vector, n(x,t) is an unknown real valued function, γ,β are positive constants and nonautonomous terms h1(x,t), and h2(x,t) are time-depended external forces satisfying quasiperiodicity conditions (see Assumption 1); non-linear terms f(s) and g(s) are given real-valued functions, satisfying
(9)|f(s)|⩽c1(sp/2+1),|g(s)|⩽c2(s1/2+1),0⩽s<∞,p<2.(10)|f(k)(s)|⩽c3,|g(k)(s)|⩽c4,k=1,2,
where cj are given positive constants for j=1,2,3,4.Letℑ be a topological space, and φ(s)∈ℑ is a function. The set
(11)ℋ(φ)={φ(h+s)∣h∈ℝ}¯
is called the hull of φ in ℑ, denoted by ℋ(φ). φ is translation compact (resp., translation bounded) if ℋ(φ) is compact (resp., bounded) in ℑ.We denote all the translation compact functions inLloc2(ℝ;X) by Lc2(ℝ;X); X is a Banach space. Apparently, φ∈Lc2(ℝ;X) implies that φ is translation bounded as follows:
(12)∥φ∥Lb2(ℝ;X)2=supt∈ℝ∫tt+1‍∥φ∥X2ds<∞.Assumption 1.
Fori=1,2, we suppose hi(x,t)∈C1(τ,+∞;Hper1(Ω)) and it satisfies quasiperiodicity conditions; namely,
(13)hi(x,t)=Hi(x,ωi(t))=Hi(x,ω1i(t),…,ωkii(t)),
and for all j∈{1,2,…,ki},
(14)Hi(x,ω1i,ω2i,…,ωji+2π,…,ωkii)=Hi(x,ω1i,ω2i,…,ωji,…,ωkii),
where ωji=αjit, 1⩽ki<∞, and {αji}j=1ki are rational and independent; Hi is differentiable to each position and
(15)Hi,∂Hi∂ωji∈Hper1(Ω).Ifhi(x,t) satisfy Assumption 1, we can consider the symbol space ℋ(hi) as(16)ℋ(hi)={Hi(α1it+ω01i,α2it+ω02i,…,αkiit+ω0kii)∣(ω01i,ω02i,…,ω0kii)=:ω0i∈Tki}¯.Since there is a continuous mappingTki→ℋ(hi):ω0i→Hi(αit+ω0i), from [18] we know that the symbol space Σ0=ℋ(h1)×ℋ(h2) can be replaced by Tk:=Tk1×Tk2. And, for each ω0:=(ω01,ω02)∈Tk, the translation operator acting on Tk can be defined as
(17)T(h):T(h)ω0=[ω0+αh]=(ω0+αh)mod(2π)k.
Therefore, Tk is translation compact.Proposition 2.
Under Assumption1, we can deduce the following properties:(i)
hi(x,t)  is translation bounded in  Hper1(Ω);that is,(18)supt⩾τ∫tt+1‍∥hi(s)∥Hper12ds⩽∫ττ+T‍∥hi(s)∥Hper12ds⩽C,whereT=max{1,2π·∏j=1kiαji},C=T·maxτ⩽t⩽τ+T∥hi(t)∥Hper12,(ii)
for ally1∈ℋ(h1),∥y1∥Lb2(ℝ;X)2⩽∥h1∥Lb2(ℝ;X)2,(iii)
hit∈Hper1(Ω),
which can be seen directly from (15) and the fact that(19)hit=∑j=1ki‍∂Hi∂ωjiαji.Moreover, hit is translation bounded in Hper1(Ω). Similarly to (18), by the continuity of hit, we can find a constant C, which is independent of t, such that
(20)∥hit∥Lb∞(τ,∞;Hper1)⩽C.For brevity, we setW(x,t):=(u(x,t),n(x,t)), Y0(x,t):=(h1(x,t),h2(x,t)), and let E0:=Hper2(Ω)×Hper1(Ω) with the norm
(21)∥W∥E0={∥u∥H22+∥n∥H12}1/2.
Similarly, we let Σ0:=ℋ(h1)×ℋ(h2) and for each Y=(y1,y2)∈Σ0,
(22)∥Y∥Σ0={∥y1∥H12+∥y2∥H12}1/2.
Then systems (5)–(8) can be rewritten as
(23)∂tW=Aσ(t)W,W|t=τ=Wτ,W(x+D,t)=W(x-D,t),
where the symbol σ(s)=Y(x,s) or ω(s) and the symbol space Σ=Σ0 or Tk.


## 3. Uniform a Prior Estimates of the Solutions

In this section, we derive uniforma priori estimates of the solutions both in time t and in symbols Y(x,t)=(y1,y2) which come from the symbol space Σ=ℋ(h1)×ℋ(h2). First we recall the following interpolation inequality.Lemma 3.
Letj,m∈ℕ∪{0}, q,r∈ℝ+, such that 0⩽j<m, 1⩽q, r⩽∞. Then we have
(24)∥Dju∥p⩽C∥Dmu∥ra∥u∥q1-a,
for u∈Wm,r(Ω)∩Lq(Ω), where Ω⊂ℝ1 and j/m⩽a⩽1, 1/p=j+a((1/r)-m)+((1-a)/q).Lemma 4.
Let Assumption1 hold. If uτ(x)∈Lper2(Ω) and Y(x,t)∈Σ, then the solutions of problem (5)–(8) satisfy
(25)∥u(t)∥⩽C1,∀t⩾t1,
where C1=C(γ,h1) and t1=C(γ,h1,R), whenever ∥uτ∥⩽R.Proof.
Taking the inner product of (5) with u in Lper2(Ω) we get that
(26)(iut+uxx-nu+iγu+g(|u|2)u,u)=(y1(x,t),u).
Taking the imaginary part of (26), we obtain that
(27)12ddt∥u∥2+γ∥u∥2=Im(y1,u).
By Young inequality and Proposition 2 we have
(28)ddt∥u∥2+γ∥u∥2⩽1γ∥y1(x,t)∥Lb2(ℝ;Hper)2⩽1γ∥h1(x,t)∥Lb2(ℝ;Hper1)2.
Then by Gronwall lemma we can complete the proof.In the following, we denote by∫‍·dx=∫Ω‍·dx, which will not cause no confusions.Lemma 5.
Under assumptions of (9) and (10) and Assumption 1, if  W(τ)∈Hper1×Hper, then solutions of problems (5)–(8) satisfy
(29)∥W(t)∥Hper1×Hper2⩽C2,∀t⩾t2,
where C2=C(γ,β,f,g,Y0,h1t) and t2=C(γ,β,f,g,Y0,h1t,R), whenever ∥Wτ∥H1×H⩽R.Proof.
Taking the inner product of (5) with ut in Hper(Ω) and taking the real part, we have
(30)-12ddt∥ux∥2-12∫‍nddt|u|2dx+Re(iγu,ut)+12∫‍g(|u|2)ddt|u|2dx=Re(y1(x,t),ut).
By (6) we know that
(31)ddt∫‍nddt|u|2dx=ddt∫‍n|u|2dx-∫‍|u|2ntdx=ddt∫‍n|u|2dx+∫‍|u|2|u|x2dx+β∫‍n|u|2dx+∫‍f(|u|2)|u|2dx-∫‍y2(x,t)|u|2dx=ddt∫‍n|u|2dx+β∫‍n|u|2dx+∫‍f(|u|2)|u|2dx-∫‍y2|u|2dx,
which shows that
(32)-12ddt∥ux∥2-12(ddt∫‍n|u|2dx+β∫‍n|u|2dx0000000000000000+∫‍f(|u|2)|u|2dx-∫‍y2|u|2dx)+12ddt∫‍G(|u|2)dx+Re(iγu,ut)-ddtRe(y1,u)+Re∫‍y1t(x,t)u¯dx=0,
where G(s) is introduced by
(33)G(s)=∫0s‍g(ξ)dξ.
Taking the inner product of (5) with u in Hper(Ω) and taking the real part, we get that
(34)Re(iut,u)-∥ux∥2-∫‍n|u|2dx+∫‍g(|u|2)|u|2dx-Re(y1(x,t),u)=0.
Multiply (34) by γ and add the resulting identity to (32) to get
(35)-12ddt∥ux∥2-12ddt∫‍n|u|2dx-12(β+2γ)∫‍n|u|2dx-12∫‍f(|u|2)|u|2dx+12∫‍y2(x,t)|u|2dx+12ddt∫‍G(|u|2)dx-γ∥ux∥2+γ∫‍g(|u|2)|u|2dx-γRe(a(x,t),u)-ddtRe(y1(x,t),u)+Re∫‍y1t(x,t)u¯dx=0.
That is,
(36)ddt(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx0000+2Re∫‍y1(x,t)u¯dx)+γ(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx0000000+2Re∫‍y1(x,t)u¯dx)+γ∥ux∥2=-∫‍f(|u|2)|u|2dx+∫‍y2(x,t)|u|2dx-γ∫‍G(|u|2)dx-(γ+β)∫‍n|u|2dx+2Re∫‍y1t(x,t)u¯dx.
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 (γ,β,f,g) but also on parameters in the brackets. For all ρ>0, when t is sufficiently large, by (9), Lemmas 3 and 4 we have
(37)|-∫‍f(|u|2)|u|2dx|⩽C∫‍|u|p+2dx+C∫‍|u|2dx⩽C∫‍(|u|2p+|u|4)dx+C∥u∥2⩽C∥u∥44+C⩽C∥ux∥∥u∥3+C⩽ρ∥ux∥2+C(ρ),|∫‍y2(x,t)|u|2dx|⩽∥y2(x,t)∥Lb2(ℝ;Hper1)2+∥u∥44⩽∥h2∥Lb2(ℝ;Hper1)2+C∥ux∥⩽ρ∥ux∥2+C2(ρ).
By (9) we deduce that
(38)|G(s)|⩽23c2s3/2+c2s,∀s⩾0.
And then(39)|-γ∫Ω‍G(|u|2)dx|⩽C∫‍(|u|3+|u|2)dx⩽C∥u∥33+C∥u∥2⩽C∥ux∥1/2∥u∥5/2+C⩽ρ∥ux∥2+C3(ρ),(40)|-(γ+β)∫Ω‍n|u|2dx|⩽ρ∥n∥2+C(ρ)∥u∥44⩽ρ∥n∥2+ρ∥ux∥2+C4(ρ),(41)|2Re∫Ω‍y1t(x,t)u¯dx|⩽∥y1t∥Lc2(ℝ;Hper1)2+∥u∥2⩽C(∥h1t∥Lc2(ℝ;Hper1)2,∥u∥2).
By (36)–(41) we deduce that
(42)ddt(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx0000+2Re∫‍y1(x,t)u¯dx)+γ(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx000000+2Re∫‍y1u¯dx)+γ∥ux∥2⩽ρ∥n∥2+4ρ∥ux∥2+C(ρ)+C(∥h1t∥Lb2(ℝ;Hper1)2,∥u∥2).
Similarly we can also deduce that(43)ddt(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx0000+2Re∫‍y1u¯dx)+β(∥ux∥2+∫‍n|u|2dx-∫‍G(|u|2)dx00000000+2Re∫‍y1u¯dx)+(2γ-β)∥ux∥2⩽ρ∥n∥2+4ρ∥ux∥2+C(ρ)+C(∥h1t∥Lb2(ℝ;Hper1)2,∥u∥2).
Taking the inner product of (6) with n in Hper(Ω), we have
(44)12ddt∥n∥2+∫‍n|u|x2dx+β∥n∥2+∫‍f(|u|2)ndx-∫‍y2(x,t)ndx=0.
By (5) we get that
(45)∫‍n|u|x2dx=∫‍nuxu¯dx+∫‍nuu¯xdx=i∫‍(utu¯x-u¯tux)dx+2Re∫‍iγuu¯xdx-2Re∫‍y2(x,t)u¯xdx,(46)ddt∫‍(iuu¯x-iuxu¯)dx=i∫‍(utu¯x+uu¯xt-uxtu¯-uxu¯t)dx=i∫‍(utu¯x-uxu¯t+utu¯x-uxu¯t)dx=2i∫‍(utu¯x-u¯tux)dx.
It comes from (44)–(46) that
(47)ddt∥n∥2+ddt∫‍i(uu¯x-uxu¯)dx+2β∥n∥2+iγ∫‍(uu¯x-uxu¯)dx⩽iγ∫‍(uu¯x-uxu¯)dx-4Re∫‍iγuu¯xdx+4Re∫‍y1(x,t)u¯xdx-2∫‍f(|u|2)ndx+2∫‍y2(x,t)ndx.
Dealing with the right hand side of inequality (47), by Lemmas 3 and 4, we get
(48)|iγ∫‍(uu¯x-uxu¯)dx|⩽2γ∥u∥∥ux∥⩽ρ∥ux∥+C1(ρ),(49)|-4Re∫‍iγuu¯xdx|⩽4γ∥u∥∥ux∥⩽ρ∥ux∥+C2(ρ),(50)|-2∫‍f(|u|2)ndx|⩽C∫‍|u|p|n|dx+C∫‍|n|dx⩽12ρ∥n∥2+C(ρ)∫‍|u|2pdx+12ρ∥n∥2+C(ρ)⩽ρ∥n∥2+C(ρ)∥ux∥p-1∥u∥p+1+C(ρ)⩽ρ∥n∥2+ρ∥ux∥2+C3(ρ),(51)|4Re∫‍y2u¯xdx|⩽4∥y1∥Lb2(ℝ;Hper1)∥ux∥⩽ρ∥ux∥+C4(ρ,∥h1∥Lb2(ℝ;Hper1)),(52)|2∫‍y2ndx|⩽2∥y2∥Lb2(ℝ;Hper1)∥n∥⩽ρ∥n∥2+C5(ρ,∥h2∥Lb2(ℝ;Hper1)).
Therefore,(53)ddt(∥n∥2+i∫‍(uu¯x-uxu¯)dx)+2β∥n∥2+iγ∫‍(uu¯x-uxu¯)dx⩽2ρ∥n∥2+4ρ∥ux∥2+C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ)).
Analogously, we can also deduce that(54)ddt(∥n∥2+i∫‍(uu¯x-uxu¯)dx)+2β∥n∥2+iβ∫‍(uu¯x-uxu¯)dx⩽2ρ∥n∥2+4ρ∥ux∥2+C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ)).
Setϵ=min{γ,β}, and
(55)E=∥ux∥2+∥n∥2+∫‍n|u|2dx-∫‍G(|u|2)dx+2Re∫‍y1u¯dx+i∫‍(uu¯x-uxu¯)dx.
Then by (42) and (53) and (43) and (54) we can respectively deduce that
(56)ddtE+γE+γ∥ux∥2+β∥n∥2⩽8ρ∥ux∥2+3ρ∥n∥2+C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ)),ddtE+βE+γ∥ux∥2+β∥n∥2⩽8ρ∥ux∥2+3ρ∥n∥2+C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ)),
which shows that if we set ρ⩽min{α/8,β/3}, we can deduce that
(57)ddtE+ϵE⩽C0,∀t⩾t0,
where C0=C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ),∥h1t(x,t)∥Lb2(ℝ;Σ)). By Gronwall lemma we have that
(58)E(t)⩽E(t0)e-γ(t-t0)+C0ϵ,∀t⩾t0.
Similarly to (39), (40), (51), and (48), for t⩾t0 we have
(59)|∫‍n|u|2dx-∫‍G(|u|2)dx+2Re∫‍y1u¯dx+i∫‍(uu¯x-uxu¯)dx|⩽ρ∥n∥2+ρ∥ux∥2+C(ρ,∥h1(x,t)∥Lb2(ℝ;Σ)).
And then
(60)|E(t0)|⩽∥ux(t0)∥2+∥n(t0)∥2+|∫‍n(t0)|u(t0)|2dx-∫‍G(|u(t0)|2)dx+2Re∫‍y1(t0)u¯(t0)dx+i∫‍(u(t0)u¯x(t0)-ux(t0)u¯(t0))dx|⩽C(R),
where C(R)=C(ρ,∥Y0(x,t)∥Lb2(ℝ;Σ),∥h1t(x,t)∥Lb2(ℝ;Σ),R) when ∥Wτ∥H1×H⩽R. Then by (58) we infer that
(61)E(t)⩽C(R)e-γ(t-t0)+C0ϵ,∀t⩾t0,⩽2C0ϵ,∀t⩾t*,
where t*=inf{t∣t⩾t0andC(R)e-ϵ(t*-t0)⩽C0/ϵ}. By (55), (59) and (61) we infer that
(62)∥ux(t)∥2+∥n(t)∥2⩽ρ∥n∥2+ρ∥ux∥2+C0.
Choosing ρ=min{γ/8,β/3,1/2}, we have
(63)∥ux∥2+∥n(t)∥2⩽C(∥Y0(x,t)∥Lb2(ℝ;Σ),∥h1t(x,t)∥Lb2(ℝ;Σ)),∀t⩾t*,
which concludes the proof by using Lemma 4.Lemma 6.
Under assumptions of Lemma5, if W(τ)∈E0=Hper2(Ω)×Hper1(Ω), then solutions of problems (5)–(8) satisfy
(64)∥W(t)∥E02⩽C2,∀t⩾t3,
where C2=C(γ,β,f,g,Y0,a0t) and t3=C(γ,β,f,g,Y0,a0t,R), whenever ∥Wτ∥E0⩽R.Proof.
Taking the real part of the inner product of (5) with uxxt in Hper(Ω), we have
(65)ddt∥uxx∥2-Re∫‍nuu¯xxtdx+Re(iγu,uxxt)+Re∫‍g(|u|2)uu¯xxtdx-Re∫‍y1(x,t)u¯xxtdx=0.
By (5) and (6), we have
(66)-Re∫‍nuu¯xxtdx=-ddt∫‍Re(nuu¯xx)dx+Re∫‍ntuu¯xxdx+Re∫‍nutu¯xxdx=-ddt∫‍Re(nuu¯xx)dx-Re∫‍uu¯xx(|u|x2+βn+f(|u|2)-y2)dx+Re∫‍nu¯xx(-inu-γu+ig(|u|2)u-iy1)dx.
Since
(67)Re(iu,uxxt)=Re∫‍iuu¯txxdx=-Re∫‍iutu¯xxdx,
we know that
(68)Re(iγu,uxxt)=γ∥uxx∥2-γRe∫‍nuu¯xxdx+γRe∫‍g(|u|2)uu¯xxdx-γRe∫‍y1u¯xxdx.
Multiply (5) by u¯ and take the real part, we find that
(69)|u|t2=2Re(iuxxu¯)-2γ|u|2-2Re(iy1u¯).
Therefore,
(70)Re∫‍g(|u|2)uu¯xxtdx=-∫‍g′(|u|2)|u|x2Re(uu¯xt)dx-∫‍g(|u|2)12ddt|ux|2dx=-∫‍g′(|u|2)|u|x2Re(uu¯xt)dx-12ddt∫‍g(|u|2)|ux|2dx+∫‍g′(|u|2)|ux|2(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx.
Now we deal with (70) to get (78). Due to equalities
(71)|u|x2=2Re(uu¯x),ddtRe(uu¯x)=Re(utu¯x)+Re(uu¯xt),
we deduce that
(72)∫‍g′(|u|2)|u|x2Re(uu¯xt)dx=ddt∫‍g′(|u|2)2Re(uu¯x)Re(uu¯x)dx-∫‍g′′(|u|2)|u|t22Re(uu¯x)Re(uu¯x)dx-∫‍g′(|u|2)2Re(uu¯x)tRe(uu¯x)dx-∫‍g′(|u|2)2Re(uu¯x)Re(utu¯x)dx.
We take care of terms in (72) as follows
(73)∫‍g′′(|u|2)|u|t22Re(uu¯x)Re(uu¯x)dx=4∫‍g′′(|u|2)(Re(uu¯x))2000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx,(74)∫‍g′(|u|2)2Re(uu¯x)tRe(uu¯x)dx=∫‍g′(|u|2)2Re(utu¯x)Re(uu¯x)dx+∫‍g′(|u|2)2Re(uu¯xt)Re(uu¯x)dx=2∫‍g′(|u|2)Re(uu¯x)0000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx+∫‍g′(|u|2)|u|x2Re(uu¯xt)dx,(75)∫‍g′(|u|2)2Re(uu¯x)Re(utu¯x)dx=2∫‍g′(|u|2)Re(uu¯x)0000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx.
It follows from (72)–(75) that
(76)-∫‍g′(|u|2)|u|x2Re(uu¯xt)dx=-ddt∫‍g′(|u|2)2Re(uu¯x)Re(uu¯x)dx+4∫‍g′′(|u|2)(Re(uu¯x))2000000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx+4∫‍g′(|u|2)Re(uu¯x)000000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx+∫‍g′(|u|2)|u|x2Re(uu¯xt)dx.
And then(77)-∫‍g′(|u|2)|u|x2Re(uu¯xt)dx=-ddt∫‍g′(|u|2)Re(uu¯x)Re(uu¯x)dx+2∫‍g′′(|u|2)(Re(uu¯x))2000000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx+2∫‍g′(|u|2)Re(uu¯x)00000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx.
From (70) and (77) we have
(78)Re∫‍g(|u|2)uu¯xxtdx=-ddt∫‍g′(|u|2)Re(uu¯x)Re(uu¯x)dx-12ddt∫‍g(|u|2)|ux|2dx+2∫‍g′′(|u|2)(Re(uu¯x))2000000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx+2∫‍g′(|u|2)Re(uu¯x)000000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx+∫‍g′(|u|2)|ux|20000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx.
By (65), (66), (68), and (78) we conclude that
(79)ddt(∥uxx∥2-2Re∫‍nuu¯xx-2∫‍g′(|u|2)(Re(uu¯x))2-∫‍g(|u|2)|u|x2-2Re∫‍y1u¯xx)+2γ(∥uxx∥2-2Re∫‍nuu¯xx-2∫‍g′(|u|2)(Re(uu¯x))2-∫‍g(|u|2)|ux|2-2Re∫‍y1u¯xx)+2γ∫‍nuu¯xxdx+4γ∫‍g′(|u|2)(Re(uu¯x))2dx+2γ∫‍g(|u|2)|ux|2dx+2γRe∫‍y1u¯xxdx-2Re∫‍uu¯xx(|u|x2+βn+f(|u|2)-y2)dx+2Re∫‍nu¯xx(-inu-γu+ig(|u|2)-iy1)dx+2γRe∫‍g(|u|2)u¯uxxdx+4∫‍g′′(|u|2)(Re(uu¯x))2(Re(iuxxu¯)-γ|u|20000000000000000000000000|u|2-Re(iy1u¯))dx+4∫‍g′(|u|2)Re(uu¯x)0000000×Re(iu¯x(uxx-nu+iγu+g(|u|2)u-y1))dx+2∫‍g′(|u|2)|ux|2000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx000000+2Re∫‍y1tu¯xxdx=0,
where ∫‍·=∫‍·dx.
For later purpose, we let(80)F(u,n,y1)=-2Re∫‍nuu¯xxdx-2∫‍g′(|u|2)(Re(uu¯x))2dxF(u,n,y1)=-∫‍g(|u|2)|ux|2dx-2Re∫‍y1u¯xxdx,(81)-G(u,n,y1,y2)=2γ∫‍nuu¯xxdx+4γ∫‍g′(|u|2)(Re(uu¯x))2dx+2γ∫‍g(|u|2)|ux|2dx+2γRe∫‍y1u¯xxdx-2Re∫‍uu¯xx(|u|x2+βn+f(|u|2)-y2)dx+2Re∫‍nu¯xx(-inu-γu+ig(|u|2)-iy1)dx+2γRe∫‍g(|u|2)u¯uxxdx+4∫‍g′′(|u|2)(Re(uu¯x))2(Re(iuxxu¯)-γ|u|200000000-Re(iy1u¯))dx+4∫‍g′(|u|2)Re(uu¯x)00000000×Re(iu¯x(|u|2uxx-nu+iγu00000000000000000+g(|u|2)u-y1))dx+2∫‍g′(|u|2)|ux|200000000×(Re(iuxxu¯)-γ|u|2-Re(iy1u¯))dx+2Re∫‍y1tu¯xxdx.
Then from (79) we have
(82)ddt(∥uxx∥2+F)+2γ(∥uxx∥2+F)=G
or
(83)ddt(∥uxx∥2+F)+γ(∥uxx∥2+F)+α∥uxx∥2=G-γF.
By Lemma 5 and Agmon inequality we have
(84)∥u(t)∥H12+∥u(t)∥∞2+∥n(t)∥H2⩽2C2,∀t⩾t2.
In the following, we denote C=C(α,β,f,g,Y0,a0t). By Lemma 3 and (84) we estimate the size of |G-γF| to get
(85)ddt(∥uxx∥2+F)+γ(∥uxx∥2+F)+γ∥uxx∥2⩽C∫‍|n|2|uxx|dx+C∫‍|ux|2|uxx|dx+C∫‍|ux|2|nu|dx+C⩽C∥uxx∥∥n∥42+C∥uxx∥∥ux∥42+C∥u∥L∞∥n∥∥ux∥42+C⩽C∥uxx∥∥nx∥1/2∥n∥3/2+C∥uxx∥3/2∥ux∥3/2+C∥u∥L∞∥n∥∥uxx∥7/4∥u∥1/4+C⩽C∥uxx∥∥nx∥1/2+C∥uxx∥7/4+C⩽γ2∥uxx∥2+β2∥nx∥2+C.
Taking the inner product of (6) with nxx in Hper(Ω), we see that
(86)-12ddt∥nx∥2+∫‍|u|x2nxxdx-β∥nx∥2+∫‍f(|u|2)nxxdx-∫‍y2nxxdx=0.
Since
(87)∫‍|u|x2nxxdx=2∫‍Re(uu¯xnxx)dx=-2∫‍Re(uu¯xxnx+|ux|2nx)dx,
by (86) we can deduce that
(88)ddt∥nx∥2+4∫‍Re(uu¯xxnx)dx+4∫‍|ux|2nxdx+2β∥nx∥2+2∫‍f′(|u|2)(uxu¯+uu¯x)nxdx-2∫‍y2xnxdx=0.
From (5) we know that
(89)iutx+uxxx-nxu-nux+iγux+g′(|u|2)|u|x2u+g(|u|2)ux-y1x(x,t)=0.
Taking the real part of the inner product to (89) with uxx in Hper(Ω), we have
(90)Re∫‍iutxu¯xx-Re∫‍nxuu¯xxdx-Re∫‍nuxu¯xxdx+Re∫‍iγuxu¯xxdx+Re∫‍g′(|u|2)|u|x2uu¯xxdx+Re∫‍g(|u|2)uxu¯xxdx-Re∫‍y1xu¯xxdx=0.
Because of
(91)ddtRe∫‍iuxu¯xxdx=2Re∫‍iutxu¯xxdx,
it holds that
(92)12ddtRe∫‍iuxu¯xxdx-Re∫‍nxuu¯xxdx-Re∫‍nuxu¯xxdx+Re∫‍iγuxu¯xxdx+Re∫‍g′(|u|2)|u|x2uu¯xxdx+Re∫‍g(|u|2)uxu¯xxdx-Re∫‍y1xu¯xxdx=0.
By (92) and (88), we find that
(93)ddt∥nx∥2+2ddtRe∫‍iuxu¯xxdx+4∫‍|ux|2nxdx+2β∥nx∥2+2∫‍f′(|u|2)|u|x2nxdx-2∫‍y2xnxdx-4Re∫‍nuxu¯xxdx+4γRe∫‍iuxu¯xxdx+4Re∫‍g′(|u|2)|u|x2uu¯xxdx+4Re∫‍g(|u|2)uxu¯xxdx-4Re∫‍y1xu¯xxdx=0.
That is,
(94)ddt(∥nx∥2+2Re∫‍iuxu¯xxdx)+2β(∥nx∥2+2Re∫‍iuxu¯xxdx)=4βRe∫‍iuxu¯xxdx-4∫‍|ux|2nxdx-2∫‍f′(|u|2)|u|x2nxdx+2∫‍y2xnxdx+4Re∫‍nuxu¯xxdx-4γRe∫‍iuxu¯xxdx-4Re∫‍g′(|u|2)|u|x2uu¯xxdx-4Re∫‍g(|u|2)uxu¯xxdx+4Re∫‍y1xu¯xxdx.
For later use, we let
(95)F1(u,n)=∥nx∥2+2Re∫‍iuxu¯xxdx,(96)G1(u,n,y1,y2)=therighthandsideof(94).
Then identity (94) as being equivalent to
(97)ddtF1+2βF1=G1.
Similarly to (85) we estimate each term in (94) and then we get
(98)ddt(∥nx∥2+2Re∫‍iuxu¯xxdx)+β(∥nx∥2+2Re∫‍iuxu¯xxdx)+β∥nx∥2⩽2βRe∫‍iuxu¯xxdx+C∥uxx∥+C∥ux∥42∥nx∥+C∥nx∥+C∥ux∥∞∥n∥∥uxx∥⩽C∥uxx∥+C∥ux∥3/2∥uxx∥1/2∥nx∥+C∥nx∥+C∥u∥1/4∥uxx∥3/4∥n∥∥uxx∥⩽γ2∥uxx∥2+β2∥nx∥2+C.
Let ϵ=min{γ,β}, and
(99)E=∥uxx∥2+∥nx∥2+F+2Re∫‍iuxu¯xxdx.
By (85) and (98) we deduce that
(100)ddtE+ϵE⩽C,∀t⩾t2,
which has the same form with (57) in the proof of Lemma 5. Similarly to the study of (57) we can derive that
(101)E(t2)⩽C(R2),E(t)⩽2Cϵ,∀t⩾t2*,
where t2*=inf{t∣t⩾t2*,C(R2)e-ϵ(t2*-t0)⩽C0/ϵ} and C(R2)=C(γ,β,f,g,Y0,h1t,R2) when ∥Wτ∥H2×H1⩽R2. By (80) we deduce that
(102)|F+2Re∫‍iuxu¯xxdx|⩽2∥u∥∞∥n∥∥uxx∥+C∥u∥∞2∥ux∥2+∥u∥∞∥ux∥2+∥h1∥Lb2(ℝ;Σ)∥uxx∥+2∥ux∥∥uxx∥+C⩽C∥uxx∥+C⩽12∥uxx∥2+C,
and then by (99), (101), and (102), we deduce that
(103)∥uxx∥2+2∥nx∥2⩽C,∀t⩾t2*,
which concludes the proof by Lemma 5.To study the AIM for the system, we construct the following higher order estimate.Lemma 7.
Under the assumptions of (9) and (10) and Assumption 1, for each Wτ∈Hper3(Ω)×Hper1(Ω), solutions of (5)–(8) satisfy
(104)supτ⩽t⩽T∥uxt∥+supτ⩽t⩽T∥u∥Hper3⩽C,∀T>τ,∥ntx∥⩽C1,∀t>t0,
where C=C(T,∥uτ∥Hper3,∥nτ∥Hper1,C0),C1=C(C0).Proof.
Taking the partial derivative of (6) with respect to x, by (9) and (10) we have
(105)∥ntx∥=∥-βnx-|u|xx2-f′(|u|2)|u|x2+h2x∥⩽C∥n∥Hper1+C∥u∥Hper2+∥h2∥Lb∞(τ,∞;Hper1)⩽C1.
Taking the partial derivatives of (5) with respect to x and t, we have that
(106)iuttx+uxxxt-(nxtu+ntux+nxut+nuxt)+iγuxt+g′′(|u|2)|u|x2|u|t2u+g′(|u|2)|u|xt2u+g′(|u|2)|u|t2ux+g′(|u|2)|u|x2ut+g(|u|2)uxt=h1xt.
Taking the inner product of (106) with uxt and taking the imaginary part, we get
(107)12ddt∥uxt∥2+γ∥uxt∥2⩽∫‍|(nxtu+ntux+nxut)u¯xt|dx+∫‍|g′′(|u|2)uu¯xt000.0.000g′×(uxutu¯2+u¯xut|u|2+uxu¯t|u|2+u¯xu¯t2)|dx+∫‍|g′(|u|2)uu¯xt00000000g′×(utxu¯+u¯xut+uxu¯t+uu¯tx)|dx+∫‍|g′(|u|2)uxu¯xt(utu¯+u¯tu)|dx+∫‍|g′(|u|2)utu¯xt(uxu¯+u¯xu)|dx+∫‍|h1xtu¯xt|dx.
Since
(108)∫‍|nxutu¯xt|dx⩽∥ut∥L∞(Ω)∫‍|nxu¯xt|dx⩽∥ut∥Hper1∥n∥Hper1∥uxt∥⩽C∥uxt∥2+C,∫‍|g′(|u|2)uu¯xtutxu¯|dx⩽C∥u∥L∞(Ω)2∥uxt∥2⩽C∥uxt∥2,
dealing with each term in (107) in the same way, we deduce that
(109)ddt∥uxt∥2⩽C∥uxt∥2+C.
By Gronwall lemma and (5) it follows that
(110)supτ⩽t⩽T∥uxt∥+supτ⩽t⩽T∥u∥Hper3⩽C,∀T>τ,
where C=C(T,∥uτ∥H3,∥nτ∥H1), which completes the proof.


## 4. Unique Existence of the Solution

In this section, we show the unique existence theorem of the solutions. Since uniforma priori estimates have been established in the former section, one can readily get the existence of the solution by Galërkin’s method (see [9, 14, 16, 19]) or operator semigroup method (see [6]). We show the theorem and prove it briefly for readers’ convenience.Theorem 8.
Under assumptions of Lemma6, for each Wτ∈E0, systems (5)–(8) has a unique global solution W(x,t)∈L∞(τ,T;E0), for all T>τ.Proof.
We prove this theorem briefly by two steps.
Step
1. Existence. ByGalërkin’s method, we apply the following approximate solution
(111)Wl(x,t)=∑j=1l‍wjl(t)ηj(x)
to approach W(x,t) and the solution of the problems (5)–(8), where {ηj}j=1∞ is a orthogonal basis of H(Ω) satisfying -Δηj=λjηj(j=1,2,…). And Wl(x,t) satisfies, j=1,2,…,l,
(112)(iutl+uxxl-nlul+iαul+g(|ul|2)ul-a,ηj)=0,(ntl+βnl+|ul|x2+f(|ul|2)-b,ηj)=0,(Wl(x,τ),ηj)=(Wτ,ηj),Wl|∂Ω=0.
It is easy to see that system (112) is an initial boundary value problem of ODE for the unknown coefficients (ujl,nj1); the solution of which is known to be unique. Like [19], by the a priori estimates in Section 3 we know that {Wl}l=1∞ converges (weakly star) to a W(x,t) which solves (5)–(8).
Step
2. Uniqueness. Suppose W1 and W2 are two solutions of the problems (5)–(8). Let W=W1-W2; then W(x,t) satisfies
(113)iut+uxx-n1u1+n2u2+iαu+g(|u1|2)u1-g(|u2|2)u2=0,nt+βn+|u1|x2-|u2|x2+f(|u1|2)-f(|u22|)=0,W|t=τ=0,W|∂Ω=0,
which has nothing to do with symbols. Similarly to [19–21], we can deduce that ∥W∥=0, which concludes the proof.Similarly to [16], by Lemma 7 we note that systems (5)–(8) has an unique global smooth solution in Hper3(Ω)×Hper1(Ω), and, moreover, it follows the following lemma.Lemma 9.
Under the assumptions of Lemma7, for each Wτ∈Hper3(Ω)×Hper1(Ω), problems (5)–(8) have a unique global solution W(x,t)∈L∞(τ,T;Hper3×Hper1), for all T>τ, satisfying
(114)∥W(t)∥Hper3×Hper12⩽C,∀t⩾t1,
where C relies on the data and t1 relies on the data and R whenever ∥Wτ∥Hper3×Hper1⩽R.


## 5. Existence of the Compact Uniform Attractor

In this section, we derive the existence of the compact uniform attractor for the system applying Ball’s idea (see [16, 17]). That is, first we construct the weak uniform attractor (the convergences are taken in the sense of weak topology), and then we show that the weak uniform attractor is actually the strong one.First we recall the following facts. Each solution trajectory for systems (5)–(8) satisfie
(115)ddt(∥uxx∥2+F)+2α(∥uxx∥2+F)=G,(116)ddt(∥nx∥2+F1)+2β(∥nx∥2+F1)=G1,
where F,  G,  F1, and G1 are given by (80), (81), (95), and (96), respectively. Moreover, by the uniform boundedness and the compactness embedding Hk(Ω)↪Hk-1(Ω) (for all k∈ℤ+) we have that F, G and F1, G1 are all weakly continuous in E0×Σ.Since we have uniformly estimated the size of solutions by Lemma6 and shown the unique existence of the solution by Theorem 8, following the method of [8, 15] we have the following theorem.Theorem 10.
Under assumptions of Theorem8, the family of processes {Uσ(t,τ)}σ∈Σ generated by systems (5)–(8) is weakly (E0×Σ,E0)-continuous and it admits a weakly compact uniform attractor 𝒜Σ in E0=Hper2(Ω)×Hper1(Ω).Proof.
Since by Lemma6 we know there exists a bounded uniform absorbing set, it suffices to prove that {Uσ(t,τ)}σ∈Σ is weakly (E0×Σ,E0)-continuous and the existence of the weak uniform attractor follows.
For any fixedt1⩾τ∈ℝ, let
(117)(Wτk,σk)⇀(Wτ,σ)inE0×Σ,
we will complete the proof if we deduce that
(118)Wσk(t1)⇀Wσ(t1)inE0,
where Wσk(t1)=(uk(t1),nk(t1))=Uσk(t1,τ)Wτk, and Wσ(t1)=(u(t1),n(t1))=Uσ(t1,τ)Wτ.
By (117) and Theorem 8 we get the boundedness
(119)∥Wτk∥E0⩽C,(120)supt∈[τ,T]∥Wσk(t)∥E0⩽C.
By Agmon inequality ∥v∥∞⩽C∥v∥H1 we see that
(121)∥Wσk(t)∥∞⩽C,∀τ⩽t⩽T.
Note that
(122)iukt=-ukxx+nkuk-iγuk-g(|uk|2)uk+y1k(x,t),(123)nkt=-|uk|x2-βnk-f(|uk|2)+y2k(x,t),
and σk=(y1k(x,t),y2k(x,t))∈Σ. By (120) and (121) we find that ∂tWσk(t)∈L∞(τ,T;H) and
(124)∥∂tWσk(t)∥L∞(τ,T;H)⩽C.
Due to Theorem 8 and (124) we know that there exists W~(t)≜(u~(t),n~(t))∈L∞(τ,T;E0) and subsequences of {Wσk(t)}, which are still denoted by {Wσk(t)}, such that
(125)Wσk(t)⇀*W~(t)inL∞(τ,T;E0),(126)∂tWσk(t)⇀*∂tW~(t)inL∞(τ,T;H).
Besides, for all t1∈[τ.T], by (120) we know that there exists W0≜(u0,n0)∈E0 such that
(127)Wσk(t1)⇀W0inE0.
By (125) and a compactness embedding theorem, we claim that
(128)uk(t)⟶u~(t)stronglyinL2(0,T;H).
In the following, we will show thatW~(t) is a solution of problems (5)–(8).
For allv∈H, ∀ψ∈C0∞(τ,T), by (122) we find that
(129)∫τT‍(iukt,ψ(t)v)dt+∫τT‍(ukxx,ψ(t)v)dt-∫τT‍(nkuk,ψ(t)v)dt+∫τT‍(iγuk,ψ(t)v)dt+∫τT‍(g(|uk|2)uk,ψ(t)v)dt-∫τT‍(y1k(x,t),ψ(t)v)dt=0.
Due to
(130)∫τT‍(nkuk,ψ(t)v)dt-∫τT‍(n~u~,ψ(t)v)dt=∫τT‍((uk-u~)nk,ψ(t)v)dt+∫τT‍(u~(nk-n~),ψ(t)v)dt,
and by (121), (128), and (125)
(131)∫τT‍((uk-u~)nk,ψ(t)v)dt⩽sup0⩽t⩽T∥nk(t)∥∞∥ψ(t)v∥L2(0,T;H)×∥uk-u~∥L2(0,T;H)⟶0,∫τT‍(u~(nk-n~),ψ(t)v)dt=∫τT‍((nk-n~),ψ(t)vu~¯)dt⟶0,
we have
(132)∫τT‍(nkuk,ψ(t)v)dt⟶∫τT‍(n~u~,v)ψ(t)dt.
Taking care of other terms of (129) in similar methods and taking the limit, we have
(133)∫τT‍(iu~t,v)ψ(t)dt+∫τT‍(u~xx,v)ψ(t)dt-∫τT‍(n~u~,v)ψ(t)dt+∫τT‍(iγu~,v)ψ(t)dt+∫τT‍(g(|u~|2)u~,v)ψ(t)dt-∫τT‍(y1(x,t),v)ψ(t)dt=0.
Therefore, in the sense of distributions it holds that
(134)iu~t+u~xx-u~n~+iγu~+g(|u~|2)u~=y1(x,t),
which shows that (u~,n~,y1(t)) satisfies (5).
For allv∈H and for all ψ∈C0∞(τ,T) with ψ(T)=0, ψ(τ)=1, by (122) we find that
(135)-∫τT‍(iuk,v)ψ′(t)dt+∫τT‍(ukxx,v)ψ(t)dt-∫τT‍(nkuk,v)ψ(t)dt+∫τT‍(iγuk,v)ψ(t)dt+∫τT‍(g(|uk|2)uk,v)ψ(t)dt-∫τT‍(y1k(x,t),v)ψ(t)dt=i(uk(τ),v).
Assumption (117) implies that
(136)uk(τ)=uτk⇀uτinH.
Then taking the limit of (135), by (136) we have
(137)-∫τT‍(iu~,v)ψ′(t)dt+∫τT‍(u~xx,v)ψ(t)dt-∫τT‍(n~u~,v)ψ(t)dt+∫τT‍(iγu~,v)ψ(t)dt+∫τT‍(g(|u~|2)u~,v)ψ(t)dt-∫τT‍(y1(x,t),v)ψ(t)dt=i(uτ,v).
While from (134) we know that
(138)-∫τT‍(iu~,v)ψ′(t)dt+∫τT‍(u~xx,v)ψ(t)dt-∫τT‍(n~u~,v)ψ(t)dt+∫τT‍(iγu~,v)ψ(t)dt+∫τT‍(g(|u~|2)u~,v)ψ(t)dt-∫τT‍(y1(x,t),v)ψ(t)dt=i(u~(τ),v).
It come from (137) and (138) that
(139)(uτ,v)=(u~(τ),v),∀v∈H,
and then
(140)u~(τ)=uτ.
Equations (134) and (140) imply that
(141)u~(t)=u(t).
For all v∈H, ∀ψ∈C0∞(τ,t1), with ψ(τ)=0, ψ(t1)=1. Then repeating the procedure of the proofs of (135)–(138), by (127) we deduce that
(142)u0=u~(t1).
It comes from (127), (141), and (142) that
(143)uk(t1)⇀u(t1)inHper2(Ω).
Similarly, we can also deduce that(144)nk(t1)⇀n(t1)inHper1(Ω),
which together with (143) proves (118) and then the theorem.To prove the strong compactness of the attractor𝒜Σ, we recall the following two lemmas.Lemma 11.
Let(X,∥·∥X) be a uniform convex Banach space (particularly, a Hilbert space) and let {xk}k⩾0 be a sequence in X. If xk⇀x0 and ∥xk∥X→∥x0∥X, then xk→x0.Lemma 12.
Let{xk}k⩾0 be a sequence in B* space X. If xk⇀x0, then
(145)supk⩾1∥xk∥X<∞,∥x0∥X⩽liminfk→∞∥xk∥X.Theorem 13.
Under assumptions of Theorem8, the weak uniform attractor 𝒜Σ in Theorem 10 is actually the strong one for the system in E0.Proof.
Since a point(w,m) belongs to the weak uniform attractor 𝒜Σ if and only if there exist two sequences {wk0,mk0}k=1∞ and {tk}k=1∞ such that for all σ(t)∈Σ, it uniformly holds that
(146)Uσ(tk,τ)(wk0,mk0)⇀(w,m)inE0,k⟶∞,
where tk→∞ as k→∞. The theorem is concluded if the weak convergence is strong.
For eachS>τ fixed, since tk→∞, we can consider it as S<tk-2τ,  k∈N+. By Lemma 6 we know Uσ(tk-S,τ)(wk0,mk0) is bounded in E0, and then there exists a (v,p)∈E0 and a subsequence of Uσ(tk-S,τ)(wk0,mk0), which is still denoted by Uσ(tk-S,τ)(wk0,mk0), such that
(147)Uσ(tk-S,τ)(wk0,mk0)⇀(v,p)inE0.
Let
(148)(wk(t),mk(t))=UT(tk-S-τ)σ(t,τ)Uσ(tk-S,τ)(wk0,mk0)=Uσ(t+tk-S-τ,tk-S)Uσ(tk-S,τ)(wk0,mk0)=Uσ(t+tk-S-τ,τ)(wk0,mk0),
where T(·) is the translation operator on Σ. Since σ(t) is quasiperiodic, there exists a σ*∈Σ such that
(149)T(tk-S-τ)σ⇀*σ*inΣ.
Therefore, by (147) and (148) and the weak (E×Σ)-continuity of Uσ∈Σ(t,τ) we see that
(150)(wk(t),mk(t))⇀Uσ*(t,τ)(v,p)inE0,∀t>τ,
and by taking t=S+τ,
(151)(w,m)=Uσ*(S+τ,τ)(v,p).
From the first equality of (148) we can consider (wk(t),mk(t)) as the solution trajectory, starting at Uσ(tk-S,τ)(wk0,mk0), created by UT(tk-S-τ)σ(t,τ). Hence, by (115) and from the boundedness that ∥Uσ(tk-S,τ)wk0∥Hper22⩽C we find that
(152)∥wk(t)∥Hper22+F(wk(t),mk(t))=e-2α(t-τ)(∥Uσ(tk-S,τ)wk0∥Hper22000.00000∥Uσ(tk-S,τ)wk0∥Hper22+F(Uσ(tk-S,τ)wk0,Uσ(tk-S,τ)mk0))+∫τt‍e-2α(t-τ)G(wk(s),mk(s))ds⩽e-2α(t-τ)(C+F(Uσ(tk-S,τ)wk0,Uσ(tk-S,τ)mk0))+∫τt‍e-2α(t-τ)G(wk(s),mk(s))ds.
Since F and G are weakly continuous in E0, by taking t=S+τ in (152), from (148), (147), (150), (151), and the Lebesgue dominated convergence theorem we get that
(153)limsupk→∞∥Uσ(tk,τ)wk0∥Hper22+F(Uσ*(S+2τ,τ)(v,p))⩽e-2αS(C+F(v,p))+∫τS+τ‍e-2αSG(Uσ*(s+τ,τ)(v,p))ds.
While(w,m)=Uσ*(S+τ,τ)(v,p), we can consider (w,m) as the solution at S+τ corresponding to the initial data (v,p) and the symbol σ*. Similarly to (152) we have
(154)∥w∥Hper22+F(w,m)=e-2αS(∥v∥Hper22+F(v,p))+∫τS+τ‍e-2αSG(Uσ*(s+τ,τ)(v,p))ds.
Deducting (154) from (153), we see that
(155)limsupk→∞∥Uσ(tk,τ)wk0∥Hper22⩽∥w∥Hper22+Ce-2αS-e-2αS∥v∥Hper22⩽∥w∥Hper22+Ce-2αS.
Since S is fixed arbitrarily, let S→∞; we conclude that
(156)limsupk→∞∥Uσ(tk,τ)wk0∥Hper22⩽∥w∥Hper22.
On the other hand, by Lemma 12, the weak convergence Uσ(tk,τ)wk0⇀w implies that
(157)liminfk→∞∥Uσ(tk,τ)wk0∥Hper22⩾∥w∥Hper22.
It follows from the previous two inequalities that
(158)limk→∞∥Uσ(tk,τ)wk0∥Hper22=∥w∥Hper22.
Similarly to the previous arguments, by using (116) we can derive that
(159)limk→∞∥Uσ(tk,τ)mk0∥Hper22=∥m∥Hper22.
By (146), (158), and (159) and Lemma 11, we conclude that Uσ(tk,τ)(wk0,mk0)→(w,m) in E0, which completes the proof.Remark 14.
We remark that up to this point the quasiperiodicity of the forces is not essentially necessary. We have actually used the uniform boundedness and the weak compactness of the symbol space inH, which can be totally satisfied by translation bounded external forces. In other words, if hi(x,t)(i=1,2) are relaxed to be translation bounded: hi∈Lb2(τ,∞;Hper1(Ω)), then all the results here still hold.


## 6. Approximate Inertial Manifolds for (5)–(8)

### 6.1. Extending and Splitting the Phase Space

From Theorem13 we know the systems (5)–(8) create a family of processes {Uσ∈Σ(t,τ)}, which admit a compact uniform attractor in E0. Then from phase plane extension formula in [18], there is a semigroup {S(t)}, where
(160)S(t)(W0,ω0)=(Uω0(t,0)W0,T(t)ω0),W0∈E0,ω0∈Tk,t⩾0,
which is created by the following autonomous system:
(161)iut+uxx-nu+iγu+g(|u|2)u=H1(x,ω1(t)),(162)nt+βn+|u|x2+f(|u|2)=H2(x,ω2(t)),(163)∂ωi(t)∂t=αi(i=1,2),(164)(W(x,0),ω(x,0))=(W0(x),ω0(x)).LetB(u,n)=un be a bilinear operator: H2×H1→H1, H(u)=|u|x2 a nonlinear operator: H2→H1, and A=-∂xx. Equations (161)–(163) can be transformed into the following abstract differential form
(165)iut-Au-B(n,u)+iγu+g(|u|2)u=H1(x,ω1(t)),nt+H(u)+βn+f(|u|2)=H2(x,ω2(t)),∂ωi(t)∂t=αi(i=1,2).
Since A is an unbounded self-conjugate compact operator, there is a complete orthogonal set {ηj}j=1∞ of eigenfunctions of A such that Aηj=λjηj, and
(166)0⩽λ1⩽λ2⩽⋯⩽λj⟶+∞,j⟶+∞.For allm>0, we let P:H→Hm:=span{η1,…,ηm} be a projective operator and let Q:=I-P:H→Hm⊥. Taking the projection of (165) we get
(167)idydt-Ay-PB(u,n)+iγy+P(g(|u|2)u)=PH1(x,ω1(x)),(168)idzdt-Az-QB(u,n)+iγz+Q(g(|u|2)u)=QH1(x,ω1(x)),(169)dpdt+PH(u)+βp+Pf(|u|2)=PH2(x,ω2(t)),(170)dqdt+QH(u)+βq+Qf(|u|2)=QH2(x,ω2(t)),
where y=Pu, z=Qu, p=Pn, and q=Qn.By Parseval’s formula we can get the following proposition.Proposition 15.
For allv∈H, there is the following expansion:
(171)v=∑j=1∞‍〈v,ηj〉ηj,
and if v∈H1,
(172)∥v∥2=∑j=1∞‍|〈v,ηj〉|2,∥∇v∥2=∑j=1∞‍λj|〈v,ηj〉|2.
Moreover, because of (166), it holds that
(173)∥∇(Qv)∥2⩾λm+1∥Qv∥2,∥∇(Pv)∥2⩽λm∥Pv∥2.From Lemma9, Proposition 15 and Agmon inequality ∥v∥∞⩽∥v∥H1, we can deduce the following lemma.Lemma 16.
If assumptions of Lemma9 are satisfied, the solution (u,n) satisfies
(174)∥u∥H3,∥n∥H1,∥n∥∞,∥u∥∞⩽C,∥Az∥,∥A1/2z∥,∥z∥,∥zt∥,∥q∥,∥qt∥⩽Cλm+1-1/2,∀t⩾t1.


### 6.2. Constructing the AIM

Now we are in the position to show the AIM for (5)–(8).SetΦ:Hm×Hm×Tk→Hm⊥×Hm⊥,
(175)Φ(y,p,ω)=(ψ1,ψ2),∀(y,p,ω)∈Hm×Hm×Tk,
where (ψ1,ψ2) satisfies
(176)-Aψ1-QB(y,p)+iγψ1+Q(g(|y|2)y)=QH1(x,ω1(t)),(177)QH(y)+βψ2+Qf(|y|2)=QH2(x,ω2(t)).
Let Π1:H×H×Tk→H×H,Π2:H×H×Tk→Tk be orthogonal projection mappings.The following theorem shows thatGraph(Φ), the graph of mapping Φ, is just an AIM for the autonomous system (165) and Π1Graph(Φ) is for problems (5)–(8), which concludes this paper.Theorem 17.
Under assumptions of Lemma9, it holds that
(178)distH2×H1((W,ω),Graph(Φ))⩽Cλm+1-1/2,t⩾t1,
where Graph(Φ)=((y,p,ω),(ψ1,ψ2)) is the graph of Φ and C depends only on the data. Moreover,
(179)distH2×H1(W(t),∏1Graph(Φ))⩽Cλm+1-1/2,t⩾t1,
which shows that Π1Graph(Φ) is an approximate inertial manifold for problems (5)–(8).Proof.
From (176) and (168) we deduce that
(180)Aψ1-Az=Q(B(u,n)-B(y,p))+iγ(ψ1-z)+Q(g(|u|2)u)-Q(g(|y|2)y)-izt=Q(B(u,n)-B(y,n)+B(y,n)-B(y,p))+iγ(ψ1-z)+g(|u|2)z-izt=Q(B(z,n)+B(y,q))+iγ(ψ1-z)+g(|u|2)z-izt=Q(nz+qy)+iγ(ψ1-z)+g(|u|2)z-izt,
which implies that
(181)∥Aψ1-Az∥⩽∥n∥∞∥z∥+∥y∥∞∥q∥+γ∥ψ1-z∥+∥g(|u|2)∥∞∥z∥+∥zt∥.
Since
(182)∥ψ1-z∥⩽λm+1-1∥Aψ1-Az∥,
from (181) and Lemma 16, we get
(183)∥Aψ1-Az∥⩽Cλm+1-1/2+γλm+1-1∥Aψ1-Az∥.
Because λj→+∞(j→∞), there exists m0>0 such that for all m>m0,
(184)∥Aψ1-Az∥⩽Cλm+1-1/2,t⩾t1.
From (177) and (170) we see that
(185)βψ2-βq=Q(H(u)-H(y))+Q(f(|u|2)-f(|y|2))+qt.
While u=Pu+Qu=y⊕z,
(186)H(u)-H(y)=|u|x2-|y|x2=|z|x2,f(|u|2)-f(|y|2)=f′(|ζ|2)(|u|2-|y|2)=f′(|ζ|2)|z|2,
where ζ∈H2 and |y|2⩽|ζ|2⩽|u|2. Then from (185), we see that
(187)∥A1/2(ψ2-q)∥⩽C∥A|z|2∥+C∥A1/2(f′(|ζ|2)|z|2)∥+C∥A1/2qt∥.
By Lemma 16 we can deduce that
(188)∥A1/2(f′(|ζ|2)|z|2)∥⩽∥(A1/2f′(|ζ|2))|z|2∥+C∥A1/2|z|2∥⩽C∥|z|2A1/2|ζ|2∥+C∥zx∥∞∥z∥⩽C∥|z|2∥+C∥z∥⩽Cλm+1-1/2.
While from Lemma 6 we know that
(189)∥Az∥⩽Cλm+1-1/2,∥A1/2z∥⩽Cλm+1-1/2,∥A1/2qt∥⩽Cλm+1-1/2.
Therefore,
(190)∥A|z|2∥⩽C∥z∥∞∥Az∥+C∥A1/2z∥∞∥A1/2z∥⩽Cλm+1-1/2.
By (187), (188), (189), and (190), we get
(191)∥A1/2(ψ2-q)∥⩽Cλm+1-1/2.
Then from (184) and (191) we can conclude that
(192)distH2×H1((W,ω),Graph(Φ))⩽Cλm+1-1/2,t⩾t1.
The estimate (179) follows from
(193)dist(W(t),∏1Graph(Φ))=dist(Uω(t,τ)Wτ,∏1Graph(Φ))=dist(∏1S(t)(W0,ω0),∏1Graph(Φ))⩽dist(∏1S(t)(W0,ω0),∏1Graph(Φ))+dist(∏2S(t)(W0,ω0),∏2Graph(Φ))=dist(S(t)(W0,ω0),Graph(Φ))=dist((W,ω),Graph(Φ))⩽Cλm+1-1/2,
and we complete the proof.


## 6.1. Extending and Splitting the Phase Space

From Theorem13 we know the systems (5)–(8) create a family of processes {Uσ∈Σ(t,τ)}, which admit a compact uniform attractor in E0. Then from phase plane extension formula in [18], there is a semigroup {S(t)}, where
(160)S(t)(W0,ω0)=(Uω0(t,0)W0,T(t)ω0),W0∈E0,ω0∈Tk,t⩾0,
which is created by the following autonomous system:
(161)iut+uxx-nu+iγu+g(|u|2)u=H1(x,ω1(t)),(162)nt+βn+|u|x2+f(|u|2)=H2(x,ω2(t)),(163)∂ωi(t)∂t=αi(i=1,2),(164)(W(x,0),ω(x,0))=(W0(x),ω0(x)).LetB(u,n)=un be a bilinear operator: H2×H1→H1, H(u)=|u|x2 a nonlinear operator: H2→H1, and A=-∂xx. Equations (161)–(163) can be transformed into the following abstract differential form
(165)iut-Au-B(n,u)+iγu+g(|u|2)u=H1(x,ω1(t)),nt+H(u)+βn+f(|u|2)=H2(x,ω2(t)),∂ωi(t)∂t=αi(i=1,2).
Since A is an unbounded self-conjugate compact operator, there is a complete orthogonal set {ηj}j=1∞ of eigenfunctions of A such that Aηj=λjηj, and
(166)0⩽λ1⩽λ2⩽⋯⩽λj⟶+∞,j⟶+∞.For allm>0, we let P:H→Hm:=span{η1,…,ηm} be a projective operator and let Q:=I-P:H→Hm⊥. Taking the projection of (165) we get
(167)idydt-Ay-PB(u,n)+iγy+P(g(|u|2)u)=PH1(x,ω1(x)),(168)idzdt-Az-QB(u,n)+iγz+Q(g(|u|2)u)=QH1(x,ω1(x)),(169)dpdt+PH(u)+βp+Pf(|u|2)=PH2(x,ω2(t)),(170)dqdt+QH(u)+βq+Qf(|u|2)=QH2(x,ω2(t)),
where y=Pu, z=Qu, p=Pn, and q=Qn.By Parseval’s formula we can get the following proposition.Proposition 15.
For allv∈H, there is the following expansion:
(171)v=∑j=1∞‍〈v,ηj〉ηj,
and if v∈H1,
(172)∥v∥2=∑j=1∞‍|〈v,ηj〉|2,∥∇v∥2=∑j=1∞‍λj|〈v,ηj〉|2.
Moreover, because of (166), it holds that
(173)∥∇(Qv)∥2⩾λm+1∥Qv∥2,∥∇(Pv)∥2⩽λm∥Pv∥2.From Lemma9, Proposition 15 and Agmon inequality ∥v∥∞⩽∥v∥H1, we can deduce the following lemma.Lemma 16.
If assumptions of Lemma9 are satisfied, the solution (u,n) satisfies
(174)∥u∥H3,∥n∥H1,∥n∥∞,∥u∥∞⩽C,∥Az∥,∥A1/2z∥,∥z∥,∥zt∥,∥q∥,∥qt∥⩽Cλm+1-1/2,∀t⩾t1.


## 6.2. Constructing the AIM

Now we are in the position to show the AIM for (5)–(8).SetΦ:Hm×Hm×Tk→Hm⊥×Hm⊥,
(175)Φ(y,p,ω)=(ψ1,ψ2),∀(y,p,ω)∈Hm×Hm×Tk,
where (ψ1,ψ2) satisfies
(176)-Aψ1-QB(y,p)+iγψ1+Q(g(|y|2)y)=QH1(x,ω1(t)),(177)QH(y)+βψ2+Qf(|y|2)=QH2(x,ω2(t)).
Let Π1:H×H×Tk→H×H,Π2:H×H×Tk→Tk be orthogonal projection mappings.The following theorem shows thatGraph(Φ), the graph of mapping Φ, is just an AIM for the autonomous system (165) and Π1Graph(Φ) is for problems (5)–(8), which concludes this paper.Theorem 17.
Under assumptions of Lemma9, it holds that
(178)distH2×H1((W,ω),Graph(Φ))⩽Cλm+1-1/2,t⩾t1,
where Graph(Φ)=((y,p,ω),(ψ1,ψ2)) is the graph of Φ and C depends only on the data. Moreover,
(179)distH2×H1(W(t),∏1Graph(Φ))⩽Cλm+1-1/2,t⩾t1,
which shows that Π1Graph(Φ) is an approximate inertial manifold for problems (5)–(8).Proof.
From (176) and (168) we deduce that
(180)Aψ1-Az=Q(B(u,n)-B(y,p))+iγ(ψ1-z)+Q(g(|u|2)u)-Q(g(|y|2)y)-izt=Q(B(u,n)-B(y,n)+B(y,n)-B(y,p))+iγ(ψ1-z)+g(|u|2)z-izt=Q(B(z,n)+B(y,q))+iγ(ψ1-z)+g(|u|2)z-izt=Q(nz+qy)+iγ(ψ1-z)+g(|u|2)z-izt,
which implies that
(181)∥Aψ1-Az∥⩽∥n∥∞∥z∥+∥y∥∞∥q∥+γ∥ψ1-z∥+∥g(|u|2)∥∞∥z∥+∥zt∥.
Since
(182)∥ψ1-z∥⩽λm+1-1∥Aψ1-Az∥,
from (181) and Lemma 16, we get
(183)∥Aψ1-Az∥⩽Cλm+1-1/2+γλm+1-1∥Aψ1-Az∥.
Because λj→+∞(j→∞), there exists m0>0 such that for all m>m0,
(184)∥Aψ1-Az∥⩽Cλm+1-1/2,t⩾t1.
From (177) and (170) we see that
(185)βψ2-βq=Q(H(u)-H(y))+Q(f(|u|2)-f(|y|2))+qt.
While u=Pu+Qu=y⊕z,
(186)H(u)-H(y)=|u|x2-|y|x2=|z|x2,f(|u|2)-f(|y|2)=f′(|ζ|2)(|u|2-|y|2)=f′(|ζ|2)|z|2,
where ζ∈H2 and |y|2⩽|ζ|2⩽|u|2. Then from (185), we see that
(187)∥A1/2(ψ2-q)∥⩽C∥A|z|2∥+C∥A1/2(f′(|ζ|2)|z|2)∥+C∥A1/2qt∥.
By Lemma 16 we can deduce that
(188)∥A1/2(f′(|ζ|2)|z|2)∥⩽∥(A1/2f′(|ζ|2))|z|2∥+C∥A1/2|z|2∥⩽C∥|z|2A1/2|ζ|2∥+C∥zx∥∞∥z∥⩽C∥|z|2∥+C∥z∥⩽Cλm+1-1/2.
While from Lemma 6 we know that
(189)∥Az∥⩽Cλm+1-1/2,∥A1/2z∥⩽Cλm+1-1/2,∥A1/2qt∥⩽Cλm+1-1/2.
Therefore,
(190)∥A|z|2∥⩽C∥z∥∞∥Az∥+C∥A1/2z∥∞∥A1/2z∥⩽Cλm+1-1/2.
By (187), (188), (189), and (190), we get
(191)∥A1/2(ψ2-q)∥⩽Cλm+1-1/2.
Then from (184) and (191) we can conclude that
(192)distH2×H1((W,ω),Graph(Φ))⩽Cλm+1-1/2,t⩾t1.
The estimate (179) follows from
(193)dist(W(t),∏1Graph(Φ))=dist(Uω(t,τ)Wτ,∏1Graph(Φ))=dist(∏1S(t)(W0,ω0),∏1Graph(Φ))⩽dist(∏1S(t)(W0,ω0),∏1Graph(Φ))+dist(∏2S(t)(W0,ω0),∏2Graph(Φ))=dist(S(t)(W0,ω0),Graph(Φ))=dist((W,ω),Graph(Φ))⩽Cλm+1-1/2,
and we complete the proof.



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*Source: 830715-2013-12-12.xml*