# Attractors of Modified Coupled Ginzburg–Landau Model

Authors: Shuhong Chen; Shuai Liu
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation-spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS-BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS-BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.


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## Body

## 1. Introduction

From 1924-1925, Bose–Einstein discovered a special class of particles that can be at any energy level at normal temperature, but if the temperature drops to near absolute zero, such particles plummet to the lowest energy level and gather together like a supersized particle, such as a sudden collapse of a large summer. This phenomenon is known as Bose–Einstein condensation (BEC) [1, 2]. Due to the peculiar nature of this phenomenon, it has attracted the interest of a wide range of experts and scholars who wish to visualize its dynamic behavior. Thus, in 2006, Machida and Koyama constructed a time-dependent Ginzburg–Landau theory, which described the crossover from a weak-coupling BCS state to a strong-coupling BEC state of superfluid atomic Fermi gases near the Feshbach resonance [3]. The coupled Ginzburg–Landau equations they constructed are as follows:(1)−idutx,t=−dg2+1U+aux,t+ga+d2v−2μφBx,t+c4m△ux,t,+g4mc−d△φBx,t−bux,t+gφBx,t2ux,t+gφBx,t,(2)iφBtx,t=−gUux,t+2v−2μφBx,t−14m△φBx,t,where t≥0, coupling coefficients a,b,c,andm are constants, μ is the chemical potential, 2v is the threshold energy of the Feshbach resonance, and d generally is a complex number with d=dr+idi.The Ginzburg–Landau equation has been particularly favored by scientists for its ability to effectively capture various types of features in the model and has yielded fruitful results [4–10]. These results are both interpretations of the existence results for the solutions [4, 5, 7–9] and explorations of the attractor problem for the Ginzburg–Landau model [6, 10]. Particularly interesting is the long-time behavior, i.e., the limiting graph of the state points moving in phase space as time changes and eventually converging, also known as attractors. The famous “butterfly effect” is the simplification of the convection equation in the atmospheric equations to obtain the trajectory shape of the attractor similar to a butterfly. The study of attractors is very interesting. In this paper, we focus on the attractor of the Ginzburg–Landau equation model from the BCS-BEC span phenomenon.Noting that the attractors can describe the long-term dynamic behavior of the system, in recent years, many scientists have turned their attention to the study of the attractor theory. However, the study of attractors is based on the existence of the weak solutions to the equations, and people have found that the Ginzburg–Landau equation is becoming more and more important. Then, more and more scientists have devoted themselves to studying the attractor of the Ginzburg–Landau equation and obtained fruitful results.Most of these results are obtained for the single Ginzburg–Landau equation. For example, the complex Ginzburg–Landau equation(3)ut=ρu+1+ivuxx−1+iμu2u.Doering CR, Gibbon J D, and Holm D D combined linearised sideband stability analysis with the theory of inertial manifolds to estimate Lyapunov and Hausdorff dimension of the attractor generated by equation (11). By replacing the last term of the above equation with 1+iμu2σu, Li and Guo showed the existence of global solution and attractor of the equation in unbounded domain by introducing weigh function [12].For the following derivative Ginzburg–Landau equation,(4)ut=α0u+α1△u+α2u2ux+α3u2uy+α4u2u¯x+α5u2u¯y−α6u2σu.When(5)3≤σ≤11+μ−vδ22/1+δ22−1.Boling and Bixiang discussed the existence of the global attractor for the equation and estimated the dimension of attractor [13]; Gao et al. improved the above results and proved the existence of the maximal attractor when(6)2<σ<11+b1δ−b62/a1δ+a62−1,and the dependence of the dimension of the attractor on the parameters on the equation is explored [14].In the following equation,(7)∂tu+vux=xu+γr+iγiuxx−βr+iβiu2u−δr+iδiu4u−λr+iλiu2ux−μr+iμiu2u¯x.When(8)γr,δr>0,4γrδr>λi−μi2.Gao and Guo obtained the existence of the solution and the global attractor of the equation and estimated the dimension of the attractor [15]; Gao proved the existence of exponential attractor [16].The results for the set of Ginzburg–Landau equations coupled by two equations are very few, and most of the coupled equations are real coefficient equations [17, 18] or consist of at least one that is a real coefficient equation. The results for the set of Ginzburg–Landau equations coupled by two equations that are both complex coefficient equations are even fewer. According to the information available to the authors, only existential results obtained by the set of equations (1) and (2) considered in this paper under specific conditions were found [5, 19]. To further explore the properties of the coupled Ginzburg–Landau system of equations (1) and (2), the authors further analysed the overall attractor problem for the system of equations (1) and (2) under specific initial value conditions [10]. Due to the special structure of the system of equations (1) and (2), the attractor results cannot be obtained according to the existing research methods on the attractor problem if appropriate conditions are not given. For this reason, Fang [6] et al. added appropriate viscosity terms to the system of equations (1) and (2) to achieve the goal that attractor theory can be established using the existing research methods on attractors. Subsequently, Jiang et al. [20] generalized the results of Fang [6] by correcting the energy inequality.In other words, by adding−iγφBx,t to equation (2) and additionally adding the external force terms fx and to equations (1) and (2), respectively, and then using the trajectory decomposition method, Jiang et al. studied the existence of a global attractor and estimated the fractal dimension of the global attractor [20].Inspired by the above researches, we consider the global attractor problem of the modified time-dependent Ginzburg–Landau equations with the damped viscous term as the following:(9)−idutx,t=−dg2+1U+aux,t+ga+d2v−2μφBx,t+c4m△ux,t,+g4mc−d△φBx,t−bux,t+gφBx,t2ux,t+gφBx,t−idfx,t,(10)iφBtx,t=−iγφBx,t−gUux,t+2v−2μφBx,t−14m△φBx,t+ihx,t,(11)ux,0=u0x,φBx,0=φB0x,x∈Ω,(12)ux,t=0,φBx,t=0,on∂Ω×0,∞,where Ω is a bounded region of Rn, t≥0, coupling coefficients a,b,c,andm are all real constants, μ is the chemical potential, 2v is the threshold energy of the Feshbach resonance, γ>0 is a damping parameter, d is generally the complex number with d=dr+idi, d2=dr2+di2, the external force terms fx,t and hx,t are real-valued functions, and its are uniformly bounded with respect to t.Compared with the above results, our result has the following features:(1)
The external force termsfx,t and hx,t are not only related to the space variable x but also related to the time variable t. This means that when dealing with the external force term, we should not only solve the difficulties caused by the space variable x but also deal with the troubles caused by the time variable t.(2)
The value range of damping coefficientγ here is larger, and the result obtained here is also broader.(3)
In particular, the resonance coefficient of Feshbach resonance in the equations considered in this paper is less than 0, which means the usual method to obtain the energy inequalities is invalidate. In order to overcome these difficulties, this paper focuses on the energy inequalities for∇u22 and ∇φB22 first and then using Poincare’s theory to deduce the estimate for u22 and finally we obtain the following result:Theorem 1.
assume thatux,t and φBx,t are the global weak solutions of equations (9)–(12), 1/U−a>0, m>0, c>0, b<0, γ>0, di>0, 3di2≤dr2, fx,t∈W21,1Q, hx,t∈W21,1Q.Then, equations (9)–(12) exist a global attractor A, such that(1)
StA=A,∀t∈R+;(2)
limt⟶∞distStB,A=0, for any bound set B⊂H1Ω, where(13)distStB,A=supx∈Binfy∈AStx−yE.AndSt is a semigroup operator generated by the weak solutions of the initial boundary problem equations (9)–(12), with(14)A=∩τ≥0∪t≥τStA¯.Finally, we get the distribution of this paper. This paper is composed of five parts. The first part is the introduction: it introduces the origin of the model, the background of the attractor research, the main advantages of the problem studied in this paper, and the main results of this paper. The second part is the basic lemmas: it introduces the basic definitions and basic lemmas used in the proof of the paper. The third part is the a priori estimation: it proves the various energy inequalities needed to establish the main results of this paper. The fourth part is the proof of the attractor: it is mainly the proof of the existence of the overall attractor. The fifth part is the conclusion: it mainly summarizes the main results obtained in this paper and presents the problems to be considered in the future.


## 2. Fundamental Lemma

In this section, we mainly introduce some basic inequalities and lemmas needed in the proof of the main result.Lemma 1 (Gronwall’s inequality [21]).
Letyt be a non-negative continuous differentiable function, for some positive constant γ, and t∈0,T, if(15)y′t+γyt≤βt.Then,(16)yt≤y0e−γt+∫0tβse−γt−sds.Definition 1 (see [22]).
We assume thatX is a complete metric space and is a continuous semigroup on X, we call the subset A of X is a global attractor of Stt≥0, if A satisfies the following conditions:
Compactness:A is a compact set.
Invariance:StA=A, ∀t≥0.
Attractiveness: For any bounded setB in X, we have(17)distStB,A⟶0,t⟶∞,where the distA,B was defined as(18)distA,B=supx∈Ainfy∈Bdistx,y.Lemma 2 (see [23]).
LetE be a Banach space, St,t≥0 is a set of semigroup operators, St:E⟶E, it satisfies St⋅Sτ=St+τ, S0=I, where I is an identity operator, and assume that the semigroup operator St satisfies the following conditions:(1)
OperatorSt is uniformly bounded on E. In other words, for any R≥0, there exist a constant CR, such that when u⟶E≤R, for any t∈0,∞, we have Stu⟶E≤CR.(2)
There exists a bounded absorbing setB0 on E, namely, for any bounded set B⊂E, there exists a constant T, such that when t≥T, we have StB⊂B0.(3)
Whent>0, St is a completely continuous operator.Then, the semigroup operator aa!37BE! % MathType! End!1!1! has a compact global attractor.


## 3. Priori Estimate

In order to prove Theorem1, we need to establish some suitable priori estimates for global weak solutions ux,t and φBx,t to equations (9)–(12). To simplify, we write Q=0,+∞;Ω and convert the equations (9) and (10) to the following equations (19) and (20).(19)utx,t+gφBtx,t,=iad−idUux,t+gφBx,t+igdU−γgφBx,t+ic4md△ux,t+gφBx,t,−ibdux,t+gφBx,t2ux,t+gφBx,t+fx,t+ghx,t,(20)φBtx,t=−γφBx,t+igUux,t+gφBx,t−ig2UφBx,t−i2v−2μφBx,t+i4m△φBx,t+hx,t.In order to establish the attractor theory of the initial boundary value problem equations (9)–(12), we should give the existence theorem of the weak solutions to the problem equations (9)–(12) first.Theorem 2.
assume that1/U−a>0, m>0, c>0, b<0, γ>0, di>0, 3di2≤dr2, fx,t∈W21,10,+∞;H1Ω, hx,t∈W21,10,+∞;H1Ω, u0x∈H1Ω, and φB0x∈H1Ω. Then, there exists a pair of global weak solutions to equations (9)–(12), and(21)ux,t∈L∞0,+∞;H1Ω,φBx,t∈L∞0,+∞;H1Ω;utx,t∈L∞0,+∞;H1Ω,φBtx,t∈L∞0,+∞;H1Ω.Theorem2 can be proved by the standard Faedo–Galerkin method. Then, the proof is omitted here.With the existence of weak solutions, we can proceed to establish some suitable energy inequalities for weak solutions. Since in this paper, the scattering direction coefficientb<0, which make us cannot obtain the appropriate energy inequalities by the conventional method. For example, we cannot get the estimation of the norm for the weak solutions directly. To overcome this difficulty, we do this in reverse and estimate the norm of the weak solution’s gradient at first and then extrapolate the properties of the weak solutions by the estimation of the gradient of the weak solutions. After then, we get the following result.Lemma 3.
assume thatux,t and φBx,t are global weak solutions to equations (9)–(12), 1/U−a>0, m>0, c>0, b<0, γ>0, di>0, 3di2≤dr2, fx,t∈W21,10,+∞;H1Ω, and hx,t∈W21,10,+∞;H1Ω.Then, there exist some positive constantsc4, c5, and M1,M2,M3,andM4, which are independent on t, such that(22)ux,tL2Ω2≤2λ21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c5λ21+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M1<+∞,φBx,tL2Ω2≤λ3∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+λ3c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M2<+∞,∇ux,tL2Ω2≤21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c51+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M3<+∞,∇φBx,tL2Ω2≤∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M4<+∞.where λ2,λ3 are Poincare’s coefficients.Proof making the inner product of equation (19) with −△ux,t+gφBx,t¯, integrating by parts, and then taking the real parts of the result equation, with d=dr+idi, we can find that(23)ddt∇ux,t+gφBx,tL2Ω2+2did21U−a∇ux,t+gφBx,tL2Ω2,=2ReigdU−γg∫∇φBx,t⋅∇ux,t+gφBx,t¯dx−cdi2md2△ux,t+gφBx,tL2Ω2,+Re−2ibd∫ux,t+gφBx,t2ux,t+gφBx,t⋅−△ux,t+gφBx,t¯dx+2Re∫∇fx,t+ghx,t⋅∇ux,t+gφBx,t¯dx.Applying Young’s inequality and Poincare’s inequality, we can get(24)ddt∇ux,t+gφBx,tL2Ω2+2did21U−a∇ux,t+gφBx,tL2Ω2,≤2ε1∇ux,t+gφBx,tL2Ω2+2Cε1∇φBx,tL2Ω2,−cdi2md2λ1∇ux,t+gφBx,tL2Ω2,+Re−2ibd∫ux,t+gφBx,t2ux,t+gφBx,t⋅−△ux,t+gφBx,t¯dx,+2ε2∇ux,t+gφBx,tL2Ω2+2Cε2∇fx,t+ghx,tL2Ω2,where λ1 is Poincare’s coefficient.Thus,(25)ddt∇ux,t+gφBx,tL2Ω2+2did21U−a−2ε1−2ε2+cdi2md2λ1∇ux,t+gφBx,tL2Ω2−2Cε1∇φBx,tL2Ω2≤Re−2ibd∫ux,t+gφBx,t|2ux,t+gφBx,t⋅−△ux,t+gφBx,t¯dx,+2Cε2∇fx,t+ghx,tL2Ω2.Note that(26)ux,t+gφBx,t2∇ux,t+gφBx,t2,=ux,t+gφBx,t⋅ux,t+gφBx,t¯⋅∇ux,t+gφBx,t⋅∇ux,t+gφBx,t¯,=ux,t+gφBx,t⋅∇ux,t+gφBx,t¯⋅ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,=∇ux,t+gφBx,t2−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,⋅∇ux,t+gφBx,t2−ux,t+gφBx,t⋅∇ux,t+gφBx,t¯=12∇ux,t+gφBx,t2+12ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,+12ux,t+gφBx,t⋅∇ux,t+gφBx,t¯−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,⋅12∇ux,t+gφBx,t2+12ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,+12ux,t+gφBx,t⋅∇ux,t+gφBx,t¯−ux,t+gφBx,t⋅∇ux,t+gφBx,t¯,=14∇ux,t+gφBx,t2+ux,t+gφBx,t⋅∇ux,t+gφBx,t¯,−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t⋅∇ux,t+gφBx,t2,+ux,t+gφBx,t¯⋅∇ux,t+gφBx,t−ux,t+gφBx,t⋅∇ux,t+gφBx,t¯,=14P+iQP−iQ,=14P2+Q2,where(27)P=∇ux,t+gφBx,t2,Q=−iux,t+gφBx,t⋅∇ux,t+gφBx,t¯−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t.Integrating by parts, and employing formula (37), we discover(28)Re−2ibd∫ux,t+gφBx,t2ux,t+gφBx,t⋅−△ux,t+gφBx,t¯dx,=−Re2ibdr−idid2∫ux,t+gφBx,t2∇ux,t+gφBx,t2+∇ux,t+gφBx,t2,⋅∇ux,t+gφBx,t2−ux,t+gφBx,t¯⋅∇ux,t+gφBx,tdx,=−Re2ibdr−idid2∫ux,t+gφBx,t∇ux,t+gφBx,tux,t+gφBx,tux,t+gφBx,tux,t+gφBx,t22+12ux,t+gφBx,t⋅∇ux,t+gφBx,t¯−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t,⋅∇ux,t+gφBx,t2dx,=−2bdid2∫ux,t+gφBx,t2∇ux,t+gφBx,t2dx−bdid2∫ux,t+gφBx,tux,t+gφBx,tux,t+gφBx,t22dx,+bdrd2Im∫ux,t+gφBx,t⋅∇ux,t+gφBx,t¯,−ux,t+gφBx,t¯⋅∇ux,t+gφBx,t⋅∇ux,t+gφBx,t2dx,=−2bdid2∫14P2+Q2dx−bdid2∫P2dx+bdrd2Im∫iQ⋅Pdx,=−2bdid2∫14P2+Q2dx−bdid2∫P2dx+bdrd2∫PQdx,=∫−3bdi2d2P2+bdrd2PQ−bdi2d2Q2dx,=−b2d2∫3diP2−2drPQ+diQ2dx.According to the properties of the binomial functions and letting(29)M=3di−dr−drdi,be a nonpositive definite matrix, namely, M=3di2−dr2≤0, noting that 3di2≤dr2 and b<0, we can find that(30)Re−2ibd∫ux,t+gφBx,t2ux,t+gφBx,t⋅−△ux,t+gφBx,t¯dx≤0.Substituting the above formula into equation (25), then(31)ddt∇ux,t+gφBx,tL2Ω2+2did21U−a−2ε1−2ε2+cdi2md2λ1⋅∇ux,t+gφBx,tL2Ω2−2Cε1∇φBx,tL2Ω2≤2Cε2∇fx,t+ghx,tL2Ω2.Choosingε1>0 and ε2>0 small enough, such that(32)ε1+ε2=di2d21U−a,then(33)ddt∇ux,t+gφBx,tL2Ω2+did21U−a+cdi2md2λ1⋅∇ux,t+gφBx,tL2Ω2−2Cε1∇φBx,tL2Ω2≤2Cε2∇fx,t+ghx,tL2Ω2.Similarly, making the inner product of the equation (20) with −△φ¯Bx,t, integrating by parts, and taking the real parts of the result equation, we obtain that(34)ddt∇φBx,tL2Ω2+2γ∇φBx,tL2Ω2=2ReigU∫∇ux,t+gφBx,t⋅∇φ¯Bx,tdx+2Re∫∇hx,t⋅∇φ¯Bx,tdx.Applying Young’s inequality, we can discover(35)ddt∇φBx,tL2Ω2+2γ−2ε3−2ε4∇φBx,tL2Ω2−2Cε3∇ux,t+gφBx,tL2Ω2≤2Cε4∇hx,tL2Ω2.Choosingε3>0 and ε4>0 small enough such that(36)ε3+ε4=γ2,then(37)ddt∇φBx,tL2Ω2+γ∇φBx,tL2Ω2−2Cε3∇ux,t+gφBx,tL2Ω2≤2Cε4∇hx,tL2Ω2.Now, adding3.5×k1 to 3.6×k2 (where k1 and k2 are arbitrary positive constants),(38)ddtk1∇ux,t+gφBx,tL2Ω2+k2∇φBx,tL2Ω2+dik1d21U−a+cdik12md2λ1−2Cε3k2∇ux,t+gφBx,tL2Ω2+γk2−2Cε1k1∇φBx,tL2Ω2≤2Cε2k1∇fx,t+ghx,tL2Ω2+2Cε4k2∇hx,tL2Ω2.Lett(39)2Cε1k1=γk22,2Cε3k2=k12did21U−a+cdi2md2λ1.That is,(40)k2=k1Cε1γCε3did21U−a+cdi2md2λ1,And letk1=1, then(41)k2=Cε1γCε3did21U−a+cdi2md2λ1.Thus,(42)ddt∇ux,t+gφBx,tL2Ω2+Cε1γCε3did21U−a+cdi2md2λ1⋅∇φBx,tL2Ω2+12did21U−a+cdi2md2λ1∇ux,t+gφBx,tL2Ω2+γ2Cε1γCε3did21U−a+cdi2md2λ1∇φBx,tL2Ω2≤2Cε2∇fx,t+ghx,tL2Ω2+2Cε4Cε1γCε3did21U−a+cdi2md2λ1∇hx,tL2Ω2.Choose positive constantsc1,c2, and c3 satisfying(43)c1=min1,Cε1γCε3did21U−a+cdi2md2λ1>0,c2==min12did21U−a+cdi2md2λ1,γ2Cε1γCε3did21U−a+cdi2md2λ1>0,c3==max2Cε2,2Cε4Cε1γCε3did21U−a+cdi2md2λ1>0.Then,(44)ddt∇ux,t+gφBx,tL2Ω2+∇φBx,tL2Ω2+c2c1∇ux,t+gφBx,tL2Ω2+∇φBx,tL2Ω2≤c3c1∇fx,t+ghx,tL2Ω2+∇hx,tL2Ω2.For(45)c2c1=c4>0,c3c1=c5>0,we have(46)c2c1=c4>0,c3c1=c5>0,Employing Gronwall’s inequality, we can find that(47)∇ux,t+gφBx,tL2Ω2+∇φBx,tL2Ω2≤∇u0x+gφBxL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds.Noting thatC4>0,C5>0,f∈W1,2Q, then there exist constants M3>0,M4>0 independent on t, such that(48)∇ux,t+gφBx,tL2Ω2≤∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M4¯<+∞,and(49)∇φBx,tL2Ω2≤∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤M4<+∞.This means that(50)∇ux,tL2Ω2=∇ux,t+gφBx,t−g∇φBx,tL2Ω2≤2∇ux,t+gφBx,tL2Ω2+2g2∇φBx,tL2Ω2≤21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c51+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤21+g2M4¯=M3<+∞.Using Poincare’s inequality, we have(51)1λ2ux,tL2Ω2≤∇ux,tL2Ω2≤21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c51+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds,and(52)1λ3φBx,tL2Ω2≤∇φBx,tL2Ω2≤∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds.The proof of Lemma3 is completed.Lemma 4.
assume thatux,t and φBx,t are the global weak solutions of equations (9)–(12), 1/U−a>0, m>0, c>0, b<0, γ>0, di>0, 3di2≤dr2, and hx,t∈W21,10,+∞;Ω.Then, there exist positive constantsc12 and c13, such that(53)utx,tL2Ω2≤21+g2u0x+gφB0xtL2Ω2+φB0txL2Ω2e−c12t+2c131+g2∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds,φBtx,tL2Ω2≤u0x+gφB0xtL2Ω2+φB0txL2Ω2e−c12t+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds.Proof taking the derivative of equation (19) with respect to t and then multiplyingux,t+gφBx,t¯t on both sides, integrating by parts, we obtain(54)∫ux,t+gφBx,ttt⋅ux,t+gφBx,t¯tdx=iad−idUux,t+gφBx,ttL2Ω2+igdU−γg∫φBtx,t⋅ux,t+gφBx,t¯tdx−ic4md∇ux,t+gφBx,ttL2Ω2−ibd∫2ux,t+gφBx,t2ux,t+gφBx,tt2dx−ibd∫ux,t+gφBx,t2⋅ux,t+gφBx,t¯t2dx+∫fx,t+ghx,tt⋅ux,t+gφBx,t¯tdx.Taking the real parts of the former equality and noting that(55)−cdi2md2≤0,we have(56)ddtux,t+gφBx,ttL2Ω2≤2adid2−2did2Uux,t+gφBx,ttL2Ω2+2ReigdU−γg∫φBtx,t⋅ux,t+gφBx,t¯tdx−Re4ibd∫ux,t+gφBx,t2ux,t+gφBx,tt2dx−Re2ibd∫ux,t+gφBx,t2⋅ux,t+gφBx,t¯t2dx+2Re∫fx,t+ghx,tt⋅ux,t+gφBx,t¯tdx.Applying Young’s inequality, we discover(57)ddtux,t+gφBx,ttL2Ω2≤2adid2−2did2Uux,t+gφBx,ttL2Ω2+2ε5ux,t+gφBx,ttL2Ω2+2Cε5φBtx,tL2Ω2+2b2di+dd2∫ux,t+gφBx,t2ux,t+gφBx,tt2dx+2ε6ux,t+gφBx,ttL2Ω2+2Cε6fx,t+ghx,ttL2Ω2.By Theorem2, we can find that(58)∫ux,t+gφBx,t2ux,t+gφBx,tt2dx≤ux,t+gφBx,tL∞Ω2ux,t+gφBx,ttL2Ω2.Combining Galiardo–Nirenberg’s inequality and Poincare inequality with Lemma3, we have(59)ux,t+gφBx,tL∞Ω≤c6ux,t+gφBx,tH1Ω1/2ux,t+gφBx,tL2Ω1/2=c6ux,t+gφBx,tL2Ω2+∇ux,t+gφBx,tL2Ω21/4⋅ux,t+gφBx,tL2Ω1/2≤c6λ+1∇ux,t+gφBx,tL2Ω21/4⋅λ1/2∇ux,t+gφBx,tL2Ω1/2=c6λ1/2λ+11/4∇ux,t+gφBx,t=c6λ1/2λ+11/4M31/2=c7<+∞,where c6, c7>0 are bounded constants and λ is Poincare’s coefficient.Let(60)c8=−2b2di+dd2c72>0,and substituting these estimates into the inequality equation (57), we can find that(61)ddtux,t+gφBx,ttL2Ω2+2did2U−2adid2−2ε5−2ε6−c8⋅ux,t+gφBx,ttL2Ω2−2Cε5φBtx,tL2Ω2≤2Cε6fx,t+ghx,ttL2Ω2.Similarly, taking the derivative of equation (20) with respect to t and then multiplying φ¯Btx,t on both sides, integrating by parts, we have(62)∫φBttx,t⋅φ¯Btx,tdx+γφBtx,tL2Ω2=−ig2U+i2v−2μφBtx,tL2Ω2+igU∫ux,t+gφBx,tt⋅φ¯Btx,tdx−i4m∇φBtx,tL2Ω2+∫htx,t⋅φ¯Btx,tdx.Taking the real parts of the former equation and applying Young’s inequality, we can get that(63)ddtφBtx,tL2Ω2+2γφBtx,tL2Ω2=Re2igU∫ux,t+gφBx,tt⋅φ¯Btx,tdx+2Re∫htx,t⋅φ¯Btx,tdx≤2ε7φBtx,tL2Ω2+2Cε7ux,t+gφBx,ttL2Ω2+2ε8φBtx,tL2Ω2+2Cε8htx,tL2Ω2.Thus,(64)ddtφBtx,tL2Ω2+2γ−2ε7−2ε8φBtx,tL2Ω2−2Cε7ux,t+gφBx,ttL2Ω2≤2Cε8htx,tL2Ω2.Now, adding3.9×k3 to 3.10×k4 (where k3 and k4 are arbitrary positive constants), we have(65)ddtk3ux,t+gφBx,ttL2Ω2+k4φBtx,tL2Ω2+2dik3d2U−2adik3d2−2ε5k3−2ε6k3−c8k3−2k4Cε7ux,t+gφBx,ttL2Ω2+2γk4−2ε7k4−2ε8k4−2k3Cε5φBtx,tL2Ω2≤2k3Cε6fx,t+ghx,ttL2Ω2+2k4Cε8htx,tL2Ω2.Choosing suitablek3, k4>0, such that c8k3≤2k4Cε7, hence(66)ddtk3ux,t+gφBx,ttL2Ω2+k4φBtx,tL2Ω2+2dik3d2U−2adik3d2−2ε5k3−2ε6k3−4k4Cε7ux,t+gφBx,ttL2Ω2+2γk4−2ε7k4−2ε8k4−2k3Cε5φBtx,tL2Ω2≤2k3Cε6fx,t+ghx,ttL2Ω2+2k4Cε8htx,tL2Ω2.Let(67)2ε5k3+2ε6k3+4k4Cε7=dik3d2U−adik3d2,2ε7k4+2ε8k4+2k3Cε5=γk4.We can compute that(68)k4=k3di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7>0.Assume thatk3=1, then(69)k4=di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7>0.Substitutingk3 and k4 into the inequality equation (66), we discover that(70)ddtux,t+gφBx,ttL2Ω2+di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7⋅φBtx,tL2Ω2+di/d2U−adi/d2ux,t+gφBx,ttL2Ω2+γdi/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7φBtx,tL2Ω2≤2Cε6fx,t+ghx,ttL2Ω2+2Cε8di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7htx,tL2Ω2.Now, let(71)c9=min1,di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7>0,c10=mindid2U−adid2,γdi/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7>0,c11=max2Cε6,2Cε8di/d2U−adi/d2−2ε5−2ε6Cε52γ−2ε7−2ε8Cε7>0.We have(72)ddtux,t+gφBx,ttL2Ω2+φBtx,tL2Ω2+c10c9ux,t+gφBx,ttL2Ω2+φBtx,tL2Ω2≤c11c9fx,t+ghx,ttL2Ω2+htx,tL2Ω2.We follow define(73)c12=c10c9>0,c13=c11c9>0.Then,(74)ddtux,t+gφBx,ttL2Ω2+φBtx,tL2Ω2+c12ux,t+gφBx,ttL2Ω2+φBtx,tL2Ω2≤c13fx,t+ghx,ttL2Ω2+htx,tL2Ω2.Employing Gronwall’s inequality,(75)ux,t+gφBx,ttL2Ω2+φBtx,tL2Ω2≤utx,0+gφBtx,0tL2Ω2+φBtx,0L2Ω2e−c12t+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds,which means that(76)ux,t+gφBx,ttL2Ω2≤utx,0+gφBtx,0tL2Ω2+φBtx,0L2Ω2e−c12t+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds,and(77)φBtx,tL2Ω2≤utx,0+gφBtx,0tL2Ω2+φBtx,0L2Ω2e−c12t+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds.We further have(78)utx,tL2Ω2=ux,t+gφBx,tt−gφBtx,tL2Ω2≤2ux,t+gφBx,ttL2Ω2+2g2φBtx,tL2Ω2≤21+g2utx,0+gφBtx,0tL2Ω2+φBtx,0L2Ω2e−c12t+2c131+g2∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds.Now, we complete the proof of Lemma4.


## 4. The Existence of Global Attractor

In order to obtain the existence of the global attractor for equations (9)–(12), we need Lemma 2 to prove Theorem 1.Proof of Theorem1: In order to using Lemma 2 to prove Theorem 1, we should verify the conditions of Lemma 2 in turn. Thus, we choose Banach space E=H1Ω×H1Ω, and assume that u⇀=ux,t,φBx,tT are a pair of weak solutions of equations (9)–(12). We further define a map St:Stu⇀x,t=u⇀x,t, such that St is a mapping of E⟶E, and S0=S0u⇀x,0=u⇀x,0. Therefore, St is a semigroup operator, which is generated by the weak solutions of equations (9)–(12). Now, we can verify the conditions of Lemma 2 one by one.(1)
At first, we would demonstrate that the operatorSt is uniformly bounded on E.Using the conclusions of Lemma3, for any sphere BR⊂E with radius R=maxM1+M3,M2+M4, let B⊂BR, we find that(79)Stux,tE2=ux,tH1Ω2=ux,tL2Ω2+∇ux,tL2Ω2≤M1+M3<+∞,and(80)StφBx,tE2=φBx,tH1Ω2=φBx,tL2Ω2+∇φBx,tL2Ω2≤M2+M4<+∞.For anyR=maxM1+M3,M2+M4>0, there exists a constant CR, such that when u⟶E≤R, we have(81)Stu⟶E≤CR.Thus, the operatorSt is uniformly bounded on E.(2)
Second, we would prove that there exists a bounded absorbing set onE.According to Lemma3 and noting that the constants c4,c5>0 and independent on t, we can deduce that(82)limt⟶+∞¯ux,tL2Ω2≤limt⟶+∞¯2λ21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c5λ21+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤limt⟶+∞¯2λ21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2+2c5λ21+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2ds≤limt⟶+∞¯2λ21+g2M4=M1<+∞.Similarly,(83)limt⟶+∞¯φBx,tL2Ω2≤limt⟶+∞¯λ3∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5λ3∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤limt⟶+∞¯λ3∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2+c5λ3∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2ds=limt⟶+∞¯λ3M4<+∞,limt⟶+∞¯∇ux,tL2Ω2≤limt⟶+∞¯21+g2∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+2c51+g2∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds≤limt⟶+∞¯21+g2M4=M3<+∞,limt⟶+∞¯∇φBx,tL2Ω2≤limt⟶+∞¯∇u0x+gφB0xL2Ω2+∇φB0xL2Ω2e−c4t+c5∫0t∇fx,s+ghx,sL2Ω2+∇hx,sL2Ω2e−c4t−sds=M4<+∞,whereM1,M2,M3,andM4 are bounded positive constants, which is independent on t.Define(84)D=maxM1,M2,M3,M4,andBD=x∈Ω,x<D.Then, for any bounded setB⊂E, there exists a constant T, such that when t≥T, we have(85)Stux,t,φBx,tE2≤D,and(86)StBD⊂A¯=ux,t,φBx,t∈E:Stux,t,φBx,tE2≤D.So,StBD⊂A¯=ux,t,φBx,t∈E:Stux,t,φBx,tE2≤D is a bounded absorbing set of operator St.(3)
Finally, we would prove that whent>0, St is a completely continuous operator.According to Lemma4, we can find that there exist some constants M5 and M6 independent on t, such that when t>0, we have(87)utx,tL2Ω2≤21+g2u0x+gφB0xtL2Ω2+φB0txL2Ω2e−c12t+2c131+g2∫0tfx,s+ghx,stL2Ω2+htx,shtx,sL2Ω2e−c12t−sds≤21+g2u0x+gφB0xtL2Ω2+φB0txL2Ω2+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2ds≤21+g2M4=M5<+∞,and similarly, we also have(88)φBtx,tL2Ω2≤utx,0+gφBtx,0L2Ω2+φBtx,0L2Ω2e−c12t+c13∫0tfx,s+ghx,stL2Ω2+htx,sL2Ω2e−c12t−sds≤M6<+∞.Combining these estimates with the results of Lemmas3 and 4, we can deduce that the functions ux,t and φBx,t are completely continuous, which means that the mapping St is a completely continuous operator.Now, applying the existence theorem of attractors, Lemma2, there exists a compact global attractor of the equations (9)–(12). Theorem 1 is proved.


## 5. Conclusion

In this paper, the results on the existence of the overall attractor under the external force from the modified Ginzburg–Landau model of the BCS-BEC spanning phenomenon under appropriate conditions are established mainly by means of the Gronwall Lemma, the properties of binomial-type functions, and energy inequalities. The result contains two cases in which the external force is related to both the time variablet, the spatial variable x, and the resonance coefficient. In the next research work, we will consider the case of the existence of the modified Ginzburg–Landau equations for the system of integral attractors from the BCS-BEC spanning model under the condition of nonequilibrium states.



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