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Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials
Nonlinear Analysis 110 (2014) 33–46
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Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials✩ Tianlong Shen, Jianhua Huang ∗ College of Science, National University of Defense Technology, Changsha 410073, PR China
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Article history: Received 11 April 2014 Accepted 21 June 2014 Communicated by Enzo Mitidieri MSC: 37L55 60H15 Keywords: Coupled fractional Ginzburg–Landau equation Fractional Laplacian Commutation estimates Well-posedness Random attractor
abstract The current paper is devoted to the well-posedness and dynamics of the stochastic coupled fractional Ginzburg–Landau equation, which describes a class of nonlinear optical fiber materials with active and passive coupled cores. By the commutation estimates and Fourier– Galerkin approximation, the global existence of weak solutions and the uniqueness criterion are established. Moreover, the existence of a global attractor is shown. Finally, we consider the long-time behavior of the stochastic coupled fractional Ginzburg–Landau equation (SCFGL) with multiplicative noise, and prove the existence of a random attractor for the random dynamical system generated by the SCFGL equation. © 2014 Elsevier Ltd. All rights reserved.
1. Fractional model for nonlinear optical fiber materials Ginzburg–Landau (GL) equations are usually applied to describe a class of optical fiber materials. There has been extensive study of the GL equations (see [1–6] and reference therein). The exact homoclinic wave and soliton solution of the GL equations have been studied in [2]. Guo et al. in [5] proved the existence of a global attractor for the GL equation. The coupled Ginzburg–Landau (CGL) equations have attracted considerable attention in modeling a class of nonlinear optical fiber materials with active and passive coupled cores. There are also many papers concerning the CGL equations (see [7–9] and reference therein). The existence of the stable solutions and exponential attractors for the CGL systems has been proved in [7,9] respectively. The fractional Laplacian operator is exactly the infinitesimal generators of Lévy stable diffusion processes, and there are many fractional models that arise in plasma, flames propagation and chemical reactions in liquids, geophysical fluid dynamics and financial market et al. There have been extensive study and application of fractional differential equations including the fractional Schrödinger equation [10], fractional Landau–Lifschitz equation [11], fractional Landau–Lifschitz–Maxwell equation [12] and fractional Ginzburg–Landau (FGL) equation [13]. Pu and Guo in [13] proved the well-posedness and dynamics for the FGL equation. However, there are some limitations to this model to describe some models with some perturbations which will lead to a very large complex system. In mathematical physics, the models can be described by stochastic partial differential equations. Based on [13], the dynamics for the stochastic FGL equation with multiplicative noise has been studied in [14]. ✩ This work was partially supported by NSFC (11371367, 11101427) and fundamental research project of NUDT (JC12-02-03).
∗
Corresponding author. Tel.: +86 73184574234. E-mail addresses: [email protected] (T. Shen), [email protected] (J. Huang).
http://dx.doi.org/10.1016/j.na.2014.06.018 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
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T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
Motivated by [13,15], in the present paper, we consider the following coupled fractional Ginzburg–Landau (CFGL) equation: ut = γ1 u − (γ2 + iγ3 )(−∆)α u + (iσ1 − σ2 )|u|2 u + iv, x ∈ R, t > 0, vt = (−µ1 + iµ2 )v − (µ3 + iµ4 )(−∆)β v + iu, x ∈ R, t > 0,
(1.1)
with the initial conditions and the periodic boundary conditions: u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ R, u(x + 2π , t ) = u(x, t ), v(x + 2π , t ) = v(x, t ),
(1.2)
t > 0 , x ∈ R,
where α, β ∈ (0, 1), u and v denote the amplitude of the electromagnetic wave in a dual-core system, t denotes the time, x is the horizontal axis of the plane wave, γ2 > 0, µ2 > 0, µ3 > 0 and µ4 > 0 are dissipation coefficients and µ1 > 1, γ1 , γ3 , σ1 , σ2 are real numbers. The fractional Laplacian (−∆)α can be regarded as a pseudo differential operator with |ξ |2α and can be realized through the Fourier transform [16]:
u(ξ ), (− ∆)α u(ξ ) = |ξ |2α
(1.3) 1 2
where u is the Fourier transform of u. In what follows, we write Λ for (−∆) . However, there is a natural question: How about the dynamics for the stochastic CFGL equations? Motivated by [14], we also consider the following stochastic CFGL equation with multiplicative noise: du = [γ1 u − (γ2 + iγ3 )(−∆)α u + (iσ1 − σ2 )|u|2 u + iv]dt + β1 udW1 (t ), x ∈ R, t > 0, dv = [(−µ1 + iµ2 )v − (µ3 + iµ4 )(−∆)β v + iu]dt + β2 udW2 (t ), x ∈ R, t > 0,
(1.4)
with the same initial conditions and periodic boundary conditions to (1.1), where α, β ∈ (0, 1), γ1 > 0, γ2 > 0, β1 > 0,
µ1 > 1 +
β22
, µ3 > 0, β2 > 0, γ3 , σ1 , σ2 , µ2 and µ4 are real numbers and W1 (t ) and W2 (t ) are two-sided Wiener processes 2 on a complete probability space. The rest of the paper is organized as follows. The working function space and some basic concepts related to the random dynamical system are introduced in Section 2. We establish the well-posedness of the weak solutions for the deterministic CFGL equation in Section 3. In Section 4, we consider the long-time behavior of the solution and the existence of a global attractor is shown. Finally, a continuous random dynamical system for the stochastic CFGL equation is constructed and the existence of a random attractor is proved in Section 5. 2. Notations and preliminaries In this section, we first review some notations for the working function space. Denote H = L2per (D ) = {u|u ∈ L2 [0, 2π ], u(x + 2π , t ) = u(x, t )},
D = [0, 2π ],
W = H × H = {(u, v)|u ∈ H , v ∈ H }, with the norm
∥u∥2H = ⟨u, u∗ ⟩ =
|u|2 dx,
∥φ∥2W = ∥u∥2H + ∥v∥2H ,
D
where φ = (u, v) ∈ W . For simplicity, we use the notation ∥ · ∥ to represent the norm for space H. In what follows, we redefine some notations to the fractional derivative and fractional Sobolev space. Since u is a periodic function, it can be expressed by a Fourier series u(x) = (F −1 u)(x) :=
1 2π ξ ∈Z
u(ξ )eiξ x ,
where
u(ξ ) :=
e−iξ y u(y)dy. D
Then for s ∈ R, denote
Λs u = F −1 (|ξ |s u(ξ )).
˙ s under the norm Finally, for any s ∈ R, we define the homogeneous Sobolev space H ∥u∥H˙ s = ∥Λ u∥ = s
ξ ∈Z
|ξ | | u(ξ )| 2s
2
21
.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
35
Similarly, for any s ∈ R, we denote the inhomogeneous Sobolev space H s under the norm
∥u∥H s = ∥J s u∥ =
12 (1 + |ξ |2 )s | u(ξ )|2 . ξ ∈Z
Denote V = H α × H β = {(u, v)|u ∈ H α , v ∈ H β }, with the norm
∥φ∥2V = ∥u∥2H α + ∥v∥2H β , for any (u, v) ∈ V . Here are two lemmas about commutator estimates developed by Kato and Ponce in [17] and Kening et al. in [18], which are key technique tools to show the well-posedness of the weak solutions and dynamics for the CFGL equation. Lemma 2.1 ([17]). Assume that s > 0 and p ∈ (1, +∞). If f and g are the Schwartz class, then
∥Λs (fg ) − f Λs g ∥Lp ≤ C (∥∇ f ∥Lp1 ∥g ∥H˙ s−1,p2 + ∥f ∥H˙ s,p3 ∥g ∥Lp4 ), and
∥Λs (fg )∥Lp ≤ C (∥f ∥Lp1 ∥g ∥H˙ s,p2 + ∥f ∥H˙ s,p3 ∥g ∥Lp4 ), where p2 , p3 ∈ (1, +∞) such that 1 p
=
1 p1
+
1 p2
=
1 p3
+
1 p4
.
Lemma 2.2 ([18]). Assume that q > 1, p ∈ [q, +∞), x ∈ Rd and 1 p
+
s d
=
1 q
.
Suppose that Λs f ∈ Lq , then f ∈ Lp and there is a constant C ≥ 0 such that
∥f ∥Lp ≤ C ∥Λs f ∥Lq and if f = Λ−s g for g ∈ Lq , then
∥Λ−s g ∥Lp ≤ C ∥g ∥Lq . Finally, we recall some definitions related to the random dynamical system and random attractors. We can refer to [19] for more details. Definition 2.3. Let (Ω , F , P ) be a probability space. A measurable flow θ = {θt }t ∈R on Ω is defined as a mapping
θ : R × Ω → Ω, which is (B (R) × F ; F )-measurable and satisfies the flow property
θt θτ = θt +τ
for t , τ ∈ R,
and θ0 = idΩ . In addition, we suppose that the measure P is invariant with respect to the flow θ . Then the quadruple (Ω , F , P , θ ) is called a metric dynamical system (MDS). Definition 2.4. Let E be a complete and separable metric space. A random dynamical system (RDS) with space E carried by a metric dynamical system (Ω , F , P , θ ) is given by the mapping
ϕ : R+ × Ω × E → E , which is (B (R+ ) × F × B (E ); B (E ))-measurable and possesses the cocycle property:
ϕ(t + τ , ω, x) = ϕ(τ , θt ω, ϕ(t , ω, x)) ∀ t , τ ∈ R+ , x ∈ E , ω ∈ Ω , ϕ(0, ω, ·) = idΩ . Definition 2.5. (i) A set-valued mapping K : Ω → 2E taking value in the closed subsets of E is said to be measurable if for each x ∈ E the mapping ω → d(x, K (ω)) is measurable, where d(A, B) = sup inf d(x, y). x∈A y∈B
A measurable set-valued mapping K is called a random set.
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T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
(ii) Let A, B be random sets. A is said to attract B if d(ϕ(t , θ−t ω)B(θ−t ω), A(ω)) → 0,
as t → ∞ P-a.s.
A is said to absorb B if P-a.s. there exists an absorption time tB (ω) such that for all t ≥ tB (ω)
ϕ(t , θ−t ω)B(θ−t ω) ⊂ A(ω). (iii) The Ω -limit set of a random set K is defined by
ΩK (ω) =
ϕ(t , θ−t ω)K (θ−t ω).
T ≥0 t ≥T
Definition 2.6. A random attractor for an RDS ϕ is a compact random set A P-a.s. satisfying: (i) A is invariant, i.e., ϕ(t , ω)A(ω) = A(θt ω) for all t > 0; (ii) A attracts all the deterministic bounded sets in E. The following theorem (cf. [19, Theorem 3.11]) yields a sufficient criterion for the existence of a random attractor. Theorem 2.7. Let ϕ be an RDS and assume that there exists a compact random set K absorbing every deterministic bounded set B ⊂ E. Then the set A(ω) =
ΩB (ω),
(2.1)
B⊂E
is a random attractor for ϕ . 3. Well-posedness of the deterministic CFGL model In this section, we will prove the existence and uniqueness criteria of the weak solutions for the periodical CFGL equation (1.1) with initial data φ|t =0 = φ0 ∈ W . Theorem 3.1. For any φ0 ∈ W , there exists a weak solution for the CFGL equation (1.1) such that
φ ∈ C ([0, T ]; W ) ∩ L2 ([0, T ]; V ), and u ∈ L4 ([0, T ]; L4 (D )). To prove the theorem, we need several lemmas. Lemma 3.2. Let φ be a solution to the deterministic CFGL equation (1.1) with initial data φ0 ∈ W , then there hold
∥φ(t )∥2 ≤ e2R1 t ∥φ0 ∥2 ,
(3.1)
and
∥φ(t )∥ + 2R2 2
t
∥φ∥ τ + 2σ2 0
2 d V˙
t
0
∥u∥4L4 dτ ≤ e2R1 t ∥φ0 ∥2 ,
(3.2)
where R1 = max{γ1 + 1, 1 − µ1 } and R2 = min{γ2 , µ3 }. Proof. Multiplying the first equation and second one of systems (1.1) with u∗ and v ∗ respectively, integrating over D and taking the real part give 1 d 2 dt
∥u∥2 = γ1 ∥u∥2 − γ2 ∥Λα u∥2 − σ2 ∥u∥4L4 + Re 1
Ω
iv u∗ dx
≤ γ1 ∥u∥2 − γ2 ∥Λα u∥2 − σ2 ∥u∥4L4 + (∥u∥2 + ∥v∥2 ), 2
1 d 2 dt
1
∥v∥2 ≤ −µ1 ∥v∥2 − µ3 ∥Λβ v∥2 + (∥u∥2 + ∥v∥2 ). 2
(3.3) (3.4)
Combining (3.3) and (3.4) leads to d dt
∥φ∥2 + 2R2 ∥φ∥2V˙ + 2σ2 ∥u∥4L4 ≤ 2R1 ∥φ∥2 ,
where R1 = max{γ1 + 1, 1 − µ1 } and R2 = min{γ2 , µ3 }.
(3.5)
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
37
Getting rid of the positive terms on the left, by the Gronwall inequality we obtain
∥φ(t )∥2 ≤ e2R1 t ∥φ0 ∥2 .
(3.6)
Then integrating (3.5) over [0, t ] yields the estimate (3.2).
Lemma 3.3. Let φ be a solution for the CFGL equation (1.1), then we have
dφ dt 2
L ([0,t ],H −α ×H −β )
≤ C.
Proof. Multiplying the first equation of the system (1.1) with the test function ψ1∗ and then integrating over D ×[0, t ] lead to
t
du dt
0
t t t t ⟨iv, ψ1∗ ⟩. (iσ1 − σ2 )⟨|u|2 u, ψ1∗ ⟩ + (γ2 + iγ3 )⟨Λ2α u, ψ1∗ ⟩ + ⟨γ1 u, ψ1∗ ⟩ − , ψ1∗ = 0
0
0
0
By the Hölder inequality, we have
t ⟨γ1 u, ψ ∗ ⟩ ≤ C ∥u∥L2 ([0,t ];H ) ∥ψ1 ∥L4 ([0,t ];H α ) , 1 0 t (γ2 + iγ3 )⟨Λ2α u, ψ ∗ ⟩ ≤ C ∥u∥L2 ([0,t ];H α ) ∥ψ1 ∥L4 ([0,t ];H α ) , 1 0 t (iσ1 − σ2 )⟨|u|2 u, ψ ∗ ⟩ ≤ C ∥u∥34 ∥ψ1 ∥L4 ([0,T ];H α ) , 1 L ([0,t ];H )
(3.7)
(3.8)
(3.9)
0
t ⟨iv, ψ ∗ ⟩ ≤ C ∥v∥L2 ([0,t ];H ) ∥ψ1 ∥L4 ([0,t ];H α ) . 1
(3.10)
0
Combining (3.7)–(3.10) gives
t du ∗ , ψ 1 ≤ C ∥ψ1 ∥L4 ([0,T ];H α ) , dt
∀ψ1 ∈ L4 ([0, t ]; H α ).
(3.11)
∀ψ2 ∈ L4 ([0, t ]; H β ).
(3.12)
0
Similarly, we can obtain
t dv ∗ , ψ2 ≤ C ∥ψ2 ∥L4 ([0,t ];H β ) , dt 0
Combining (3.11) and (3.12) gives
t dφ ≤ C ∥ψ∥L4 ([0,t ];H α ×H β ) , , ψ dt
(3.13)
0
where ψ = (ψ1 , ψ2 ). Therefore,
dφ dt 2
L ([0,t ],H −α ×H −β )
dφ ≤ dt 4
L ([0,t ],H −α ×H −β )
Thus, the proof of Lemma 3.3 is completed.
≤ C.
Next it remains to show that ∥φ(t )∥W is a continuous function for t ≥ 0. Let Iψ (t ) = (⟨u(t ), ψ1∗ (t )⟩, ⟨v(t ), ψ2∗ (t )⟩). Lemma 3.4. Iψ (t ) is a continuous function of t for any ψ ∈ H × H. Proof. Multiplying the first equation and the second one of the CFGL equation (1.1) with ψ1∗ and ψ2∗ respectively, then integrating over D × [0, T ] give Iψ1 (t ) = ⟨u(t ), ψ1 ⟩ = ⟨u0 , ψ1 ⟩ + γ1
t
⟨u, ψ1 ⟩dτ − (γ2 + iγ3 ) 0 t 2 ∗ − (iσ1 − σ2 ) ⟨|u| u, ψ1 ⟩dτ + i ⟨v, ψ1∗ ⟩dτ , ∗
∗
∗
t
⟨Λ2α u, ψ1∗ ⟩dτ
0
0
Iψ2 (t ) = ⟨v(t ), ψ2∗ ⟩
= ⟨v0 , ψ2∗ ⟩ + (−µ1 + iµ2 )
0
t
⟨v, ψ2∗ ⟩dτ − (µ3 + iµ4 )
t
0
⟨Λ2β v, ψ2∗ ⟩dτ + i
t
⟨u, ψ2∗ ⟩dτ . 0
38
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
First, let ψ = (ψ1∗ , ψ2∗ ) ∈ C ∞ (D ) × C ∞ (D ) and 0 ≤ t1 < t2 ≤ T . According to [13], we have 1
|Iψ1 (t2 ) − Iψ1 (t1 )| ≤ (|γ1 | ∥ψ1 ∥L∞ + |γ2 + iγ3 | ∥Λ2α ψ1∗ ∥L∞ )∥u∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 + |iσ1 − σ2 | ∥ψ1 ∥L∞ ∥u∥3L4 ([t
1
4 1 ,t2 ];L )
1
|t1 − t2 | 4 + |i| ∥ψ1 ∥L∞ ∥v∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 ,
(3.14)
and 1
|Iψ2 (t2 ) − Iψ2 (t1 )| ≤ (| − µ1 + iµ2 | · ∥ψ2 ∥L∞ + |µ3 + iµ4 | · ∥Λ2β ψ2 ∥L∞ )∥v∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 1
+ ∥ψ2 ∥L∞ ∥u∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 .
(3.15)
Combining (3.14)–(3.15) gives
|Iψ (t2 ) − Iψ (t1 )| ≤ |Iψ1 (t2 ) − Iψ1 (t1 )| + |Iψ2 (t2 ) − Iψ2 (t1 )| √ 1 ≤ Cψ 2|φ|L2 ([t1 ,t2 ];W ) |t1 − t2 | 2 + Cψ ∥u∥3L4 ([t √
1
3
1
4 1 ,t2 ];L ) 3
|t1 − t2 | 4 1
2 2Cψ eR1 T ∥φ0 ∥2W |t1 − t2 | 2 + Cψ e 2 R1 T ∥φ0 ∥W |t1 − t2 | 4 .
≤
(3.16)
Therefore, the continuity of Iψ (t ) follows. Next we use a density argument to extend to the general case for ψ ∈ H × H. Let ϵ > 0 be an arbitrary positive number, and for ψ ∈ H × H, we may select some ψϵ ∈ C ∞ (D ) × C ∞ (D ) such that ∥ψϵ − ψ∥W ≤ ϵ . By the triangle inequality and the Hölder inequality, we have
|Iψ (t2 ) − Iψ (t1 )| ≤ |⟨φ(t2 ), ψ⟩ − ⟨φ(t2 ), ψϵ ⟩| + |⟨φ(t1 ), ψϵ ⟩ − ⟨φ(t1 ), ψ⟩| + |⟨φ(t2 ), ψϵ ⟩ − ⟨φ(t1 ), ψϵ ⟩| ≤ ϵ(∥φ(t1 )∥W + ∥φ(t2 )∥W ) + |Iψϵ (t2 ) − Iψϵ (t1 )|.
(3.17)
Since Iψϵ is continuous in t , ∥φ(t1 )∥W and ∥φ(t2 )∥W are independent of ϵ and ϵ is arbitrary, the continuity of Iψ (t ) follows for any ψ ∈ H × H. We are now in a position to prove Theorem 3.1. Proof. The existence of the weak solutions can be proved by Fourier–Galerkin approximation and compactness argument. Let {e1 , e2 , . . . , eN } be an orthogonal basis of H and PN be the orthogonal projection onto its subspace span {e1 , e2 , . . . , eN }. We construct the two abstract ODEs in this finite-dimensional space that approximate the CFGL equation
t ⟨uN , ψ1∗ ⟩dτ − (γ2 + iγ3 ) ⟨Λ2α uN , ψ1∗ ⟩dτ 0 0 t 2 ∗ − (iσ1 − σ2 ) ⟨|uN | uN , ψ1 ⟩dτ + i ⟨vN , ψ1∗ ⟩dτ ,
⟨uN (t ), ψ1∗ ⟩ − ⟨u0N , ψ1∗ ⟩ = γ1
t
(3.18)
0
t t ⟨vN (t ), ψ2∗ ⟩ − ⟨v0N , ψ2∗ ⟩ = (−µ1 + iµ2 ) ⟨vN , ψ2∗ ⟩dτ − (µ3 + iµ4 ) ⟨Λ2β vN , ψ2∗ ⟩dτ 0 0 t ∗ + i ⟨uN , ψ2 ⟩dτ ,
(3.19)
0
where ψ ∈ C ∞ (D ) × C ∞ (D ). For any fixed T > 0, Lemmas 3.2–3.4 also hold for φN , in particular
∥φN (t )∥ + 2γ2 2
t
α
∥Λ uN ∥ dτ + 2µ3 2
0
≤ ∥φ0N ∥2 + 2R1
0
t
β
∥Λ vN ∥ dτ + 2σ2 2
t
0
∥uN ∥4L4 dτ
t
∥φN ∥2 dτ .
(3.20)
0
It follows that {φN } is bounded in L2 (0, T ; V ) and {uN } is bounded in L4 (0, T ; L4 ), from which we know {φN } is weakly compact and there exists a subsequence (still denoted as {φN }) such that φN → φ weakly in L2 (0, T ; V ), also {uN } → uN weakly dφ in L4 (0, T ; L4 ). In addition, Lemmas 3.2–3.4 show {φN } is weakly compact in W for every t ≥ 0 and dtn is bounded in L2 (0, T ; −α −β 2 H × H ). Therefore, {φN } is compact in L (0, T ; W ) (thanks to Theorem 5.1 in [20]). We obtain that ⟨φN , ψ ∗ ⟩ is a continuous function for t ≥ 0 from Lemma 3.4. Since the estimates (3.16) and (3.17) are independent of N, it is equicontinuous in C ([0, T ]) for every ψ ∈ W . On the other hand, it follows from Lemma 3.2 that {⟨φN , ψ ∗ ⟩} is uniformly bounded in C [0, T ]. Therefore, {⟨φN , ψ ∗ ⟩} is compact in C [0, T ]. From the above argument, we know that ⟨φN , ψ ∗ ⟩ is a continuous function for t ≥ 0 and it converges to ⟨φ, ψ ∗ ⟩ for every t ≥ 0. From [13] we know that |u2N |uN converges weakly to |u2 |u in L1 (0, T ; L1 ) and the convergence for the linear is obvious. Therefore, the limit φ satisfies the CFGL equation (1.1) in the sense of (3.18)–(3.19).
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
39
We end this section with the proof of the uniqueness criteria of the weak solutions. Theorem 3.5. Assume that α ∈ ( 12 , 1], β ∈ (0, 1) and T > 0, there is at most one solution for the deterministic CFGL equation (1.1) such that
φ ∈ C (0, T ; W ) ∩ L2 (0, T ; V ), if 2
u ∈ L 1−θ (0, T ; L4 ),
θ=
1 4α
.
Proof. Let uA and uB be two solutions for the first term of Eq. (1.1) and vA and vB be two solutions for the second term of Eq. (1.1), then w1 = uA − uB and w2 = vA − vB satisfy
w1t = γ1 w1 − (γ2 + iγ3 )(−∆)α w1 + (iσ1 − σ2 )|u|2 u + iw2 , w2t = (−µ1 + iµ2 )w2 − (µ3 + iµ4 )(−∆)β w2 + iw1 . It follows from [13] that d dt
∥w1 ∥ ≤ C γ1 + ∥U ∥ + ∥U ∥ 2
2 L4
2 1−θ 4 L
∥w1 ∥2 + ∥w1 ∥2 + ∥w2 ∥2 ,
(3.21)
where ∥U ∥2 = ∥uA ∥2 + ∥uB ∥2 . It is easy to obtain d dt
∥w2 ∥2 ≤ −2µ1 ∥w2 ∥2 + ∥w1 ∥2 + ∥w2 ∥2 .
(3.22)
Combining (3.21)–(3.22) leads to d dt
∥ω∥
2 W
≤ C γ1 + ∥U ∥ + ∥U ∥ 2 L4
2 1−θ 4 L
+ 2 ∥w1 ∥2 + 2(1 − µ1 )∥w2 ∥2 ,
(3.23)
where w = (w1 , w2 ). Therefore, d dt
∥ω∥2W ≤ R3 ∥w∥2W ,
(3.24)
where
R3 = max C
2
γ1 + ∥U ∥2L4 + ∥U ∥L14−θ
+ 2, 2(1 − µ1 ) .
The uniqueness of the solution follows from the Gronwall inequality. Thus, the proof is completed.
4. Global attractors of the deterministic CFGL model In this section, we first establish that the solutions are globally smooth without restrictions on parameters. Theorem 4.1. Let α ∈ ( 12 , 1] and β ∈ (0, 1]. Then for all s ≥ 0, the solutions of Eq. (1.1) with initial data φ0 ∈ H s × H s obey
∥Λs φ(t )∥W ≤ C where C is a constant depending on T , ∥φ0 ∥H s ×H s and the parameters. Proof. Multiplying the first term of Eq. (1.1) with (−∆)s u∗ , then integrating over D and taking the real part, we obtain d dt
∥Λ u∥ + 2γ2 ∥Λ s
2
s+α
2 2s ∗ 2s ∗ u∥ = 2γ1 ∥Λ u∥ + 2Re (iσ1 − σ2 ) |u| uΛ u dx + 2Re i v Λ u dx . 2
s
2
According to [13], we can get the following result by the Hölder inequality and Lemmas 2.1–2.2.
2 |u|2 uΛ2s u∗ dx ≤ δ∥Λα+s ∥2 + C (δ)∥Λs u∥2 + C ∥Λs u∥ ∥u∥44 . L It follows from the Hölder inequality that
i v Λ2s u∗ dx ≤ 1 ∥Λs u∥2 + 1 ∥Λs v∥2 . 2 2
40
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
We derive that d dt
∥Λs u∥2 + 2γ2 ∥Λs+α u∥2 ≤ Cσ δ∥Λs+α u∥2 + (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 1)∥Λs u∥2 + ∥Λs v∥2 .
2 γ2 , Cσ
Let δ = d
dt
then
∥Λs u∥2 ≤ (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 1)∥Λs u∥2 + ∥Λs v∥2 .
(4.1)
Multiplying the second term of Eq. (1.1) with (−∆)s v ∗ , then integrating over D and taking the real part give d dt
∥Λs v∥2 + 2µ3 ∥Λs+β v∥2 ≤ −2µ1 ∥Λs v∥2 + ∥Λs v∥2 + ∥Λs u∥2 .
(4.2)
Combining (4.1)–(4.2) leads to d dt
∥Λs φ∥2W ≤ R4 ∥Λs φ∥2 ,
(4.3)
where R4 = max{2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 2, 2(1 − µ1 )}. Since ∥u(t )∥4L4 is integrable, applying the Gronwall inequality gives
∥Λs φ∥2W ≤ ∥Λs φ(0)∥2W e
t
0 R4 (s)ds
< ∞,
∀s ≥ 0, t ∈ [0, T ].
(4.4)
Next we consider the long-time behavior in H × H for the solutions of the CFGL equation (1.1). The basic concepts and properties of attractors can be found in [21]. We have the following theorem. Theorem 4.2. Assume that α ∈ ( 21 , 1] and β ∈ (0, 1]. Then the solution operator S : S (t )φ0 = φ(t ) for all t > 0 of the CFGL equation (1.1) well defines a semigroup in the space H × H and the following statements hold: (i) (ii) (iii) (iv)
For any t > 0, S (t ) is continuous in H × H; For any φ0 ∈ H × H, S is continuous from [0, T ] to H × H; For any t > 0, S (t ) is compact in H × H; The semigroup {S (t )}t ≥0 possesses a global attractor A in H × H.
Proof. The results in Section 3 show that the solution φ(t ) for the CFGL equation (1.1) well defines a semigroup S (t ) on H × H. First, we consider the absorbing set in H × H. By elementary estimates, we obtain 1 2
1
1
2
2
∥u∥2 + γ2 ∥Λα u∥2 + σ2 ∥u∥4L4 − γ1 ∥u∥2 − ∥u∥2 − ∥v∥2 ≤ 0,
and 1 2
1
1
2
2
∥v∥2 + µ3 ∥Λβ v∥2 + µ1 ∥v∥2 − ∥u∥2 − ∥v∥2 ≤ 0.
Thus, we obtain d dt
∥φ∥2W + 2γ2 ∥Λα u∥2 + 2µ3 ∥Λβ v∥2 − 2(γ1 + 1)∥u∥2 − 2(1 − µ1 )∥v∥2 + 2σ2 ∥u∥4L4 ≤ 0.
(Case i) If γ1 + 1 < 0, 1 − µ1 < 0, then d dt
∥φ∥2W − 2R5 ∥φ∥2W ≤ 0,
where R5 = max{γ1 + 1, 1 − µ1 } < 0. Therefore,
∥φ∥W ≤ ∥φ0 ∥W eR5 t , from which it follows that for any φ0 ∈ W ,
∥φ∥W → 0,
as t → ∞.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
41
(Case ii) If γ1 + 1 = 0, 1 − µ1 = 0, the direct result leads to
∥φ∥W ≤ ∥φ0 ∥W ,
as t → ∞.
(Case iii) If γ1 + 1 > 0, 1 − µ1 < 0, by Young’s inequality, we have 1 2
σ2 ∥u∥L4 +
4π
σ2
(γ1 + 1)2 ≥ 2(γ1 + 1)∥u∥2 ,
then d dt
∥φ∥2W + 2γ2 ∥Λα u∥2 + 2µ3 ∥Λβ v∥2 + 2(γ1 + 1)∥u∥2 + 2(µ1 − 1)∥v∥2 + σ2 ∥u∥4L4 ≤
8π (γ1 + 1)2
σ2
.
(4.5)
Therefore, d dt
8π (γ1 + 1)2
∥φ∥2W + R6 ∥φ∥2W ≤
σ2
,
where R6 = min{2(γ1 + 1), 2(µ1 − 1)} > 0. Applying the Gronwall inequality gives
∥φ∥2W ≤ e−R6 t ∥φ0 ∥2W +
8π (γ1 + 1)2
σ2
(1 − e−R6 t ) ∀t ≥ 0.
(4.6)
Therefore, lim ∥φ∥2W ≤ ρ02 ,
ρ02 =
t →∞
8π (γ1 + 1)2
σ2
.
We can infer the existence of an absorbing ball in H × H from (4.6). For any ρ > ρ0 , the balls B(0, ρ) are positively invariants for the semigroup S (t ) associated with the CFGL equation (1.1) and these balls are absorbing in H × H. Denote B0 the ball B(0, ρ0′ ) with any fixed ρ0′ > ρ0 . For any bounded set B , there exists a positive t0 = B(0, ρ0 ) for t ≥ t0 = t0 (B , ρ0 ). Integrating (4.5) from t to t + 1 leads to ′
′
∥φ(t + 1)∥2W + 2R2
t +1
∥φ∥2V˙ ds + σ2
t
8π (γ1 + 1)2
t +1
∥u∥4L4 ds ≤ ∥φ(t )∥2W +
t
σ2
1 R
log
ρ2 ρ0′2 −ρ02
such that S (t )B ⊂
.
Then it follows from (4.6) that t +1
∥φ∥2V˙ ds + σ2
2R2 t
t +1
′
∥u∥4L4 ds ≤ ρ02 +
t
8π (γ1 + 1)2
σ2
,
∀t ≥ t0 .
So for all t ≥ t0 , t +1
∥φ∥2V˙ ds ≤ a1 ,
t
t +1
t
∥u∥4L4 ds ≤ a2 ,
(4.7)
are uniformly bounded independent of φ0 . Next we consider the absorbing set in H α × H β . Similarly to the proof of Theorem 4.1, multiplying the first term and the second term of Eq. (1.1) with (−∆)α u∗ and (−∆)β v ∗ respectively, then integrating over D and taking the real parts, it follows from [13] and the Hölder inequality that d dt
∥Λα u∥2 + 2γ2 ∥Λ2α u∥2 ≤ (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 )∥Λα u∥2 +
1
γ2
∥v∥2 + (Cσ δ + γ2 )∥Λ2α u∥2 ,
(4.8)
and d dt
∥Λβ v∥2 + 2µ3 ∥Λ2β v∥2 ≤ −2µ1 ∥Λβ v∥2 + 2µ3 ∥Λ2β v∥2 +
Let δ = d dt
γ2 Cσ
1 2µ3
∥ u∥ 2 .
(4.9)
. Then combining (4.8)–(4.9) gives
∥φ∥2V˙ ≤ h1 ∥φ∥2V˙ + h2 ,
(4.10)
42
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
where 2µ1 < 0 < h1 = 2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 , and h2 = R7 ∥φ∥2W ,
R7 = max
1
,
1
2µ3 γ2
.
By (4.7), we obtain t +1
h1 (s)ds ≤ 2γ1 + Cσ C (δ) + CCσ a2 , a3 ,
t +1
′
h2 (s)ds ≤ R7 ρ02 , a4 ,
∀t ≥ t0 .
(4.11)
t
t
Applying the uniform Gronwall lemma [21] leads to d dt
∥φ∥2V˙ ≤ (a1 + a4 )ea3 ,
∀t ≥ t0 + 1.
(4.12)
The result (4.12) can assure us the existence of an absorbing ball of the solutions in H α × H β . Let φ0 ∈ B be a bounded set. Since B1 is bounded in H α × H β , and the embedding H α × H β ↩→ H × H is compact, we obtain
S (t )B is relative compact in H × H .
(4.13)
t ≥ t0 + 1
Thus, the assertion of item (iii) has been proved. Notice that items (i)–(ii) can be verified similar to one in Section 3, and item (iv) is a direct corollary of items (i)–(iii). Thus, the proof is completed. 5. Stochastic CFGL model with multiplicative noise In this section, we will establish the well-posedness and random dynamical system for the stochastic CFGL equation (1.4) with multiplicative noise of Itô form in R. Due to the special linear multiplicative noise, the stochastic CFGL equation (1.4) can be reduced to the equations with random coefficients by a suitable change of variable in the following form:
z1 (t ) = e−β1 W1 (t ) , z2 (t ) = e−β2 W2 (t ) ,
satisfy the stochastic differential equations:
1 dz1 (t ) = β12 z1 dt − β1 z1 (t )dW1 (t ), 2
dz2 (t ) = 1 β 2 z1 dt − β2 z2 (t )dW2 (t ). 2 2
We translate the unknown u′ (t ) = z1 (t )u(t ), v ′ (t ) = z2 (t )v(t ) to obtain the following random differential equations:
′ 1 2 du (t ) = γ + β u′ − (γ2 + iγ3 )(−∆)α u′ + (iσ1 − σ2 )z1−2 |u′ |2 u′ + iz1 z2−1 v ′ , 1 1 dt 2 dv ′ (t ) 1 = −µ1 + iµ2 + β22 v ′ − (µ3 + iµ4 )(−∆)β v ′ + iz1−1 z2 u′ , dt
(5.1)
2
with the initial conditions at time s and the periodic boundary conditions:
u′ (x, s) = us (x), v ′ (x, s) = vs (x), x ∈ R, u′ (x + 2π , t ) = u(x, t ), v ′ (x + 2π , t ) = v(x, t ),
t > 0, x ∈ R.
(5.2)
In what follows, we construct a random dynamical system modeling the stochastic CFGL equation (5.1). First, we consider the set of continuous functions with value 0 at 0,
Ω1 = {ω1 ∈ C (R, R) : ω1 (0) = 0},
Ω2 = {ω2 ∈ C (R, R) : ω2 (0) = 0}.
Let W1 (t , ω1 ) = ω1 (t ) and W2 (t , ω2 ) = ω2 (t ), (Ω1 , F , P) and (Ω2 , F , P) be complete probability space. A family of measure preserving and ergodic transformation of (Ω1 , F , P, (θt )t ∈R ) and (Ω2 , F , P, (θt )t ∈R ) can be defined by
θt ω1 (s) = ω1 (t + s) − ω1 (s),
θt ω2 (s) = ω2 (t + s) − ω2 (s).
For the Winner process Wj (t ), it follows that E [|Wj (t ) − Wj (s)|4 ] ≤ 3|t − s|,
j = 1, 2.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
43
j , which is a continuous function, such that P ({W j (t , ω) ̸= Due to the Kolmogorov Theorem, there exists an equal process W Wj (t , ω)}) = 0. Thus, for any fixed ω = (ω1 , ω2 ) ∈ Ω1 × Ω2 , the existence and uniqueness of the solution for the stochastic CFGL equation (1.4) can be obtained by the same method in Section 3, which defines a stochastic dynamical system (S (t , s; ω))t ≥s,ω∈Ω1 ×Ω2 by S (t , s; ω)φs = φ(t , ω; s, φs ) = φ ′ (t , ω; s, φs z (s, ω))z (t , ω),
(5.3)
where z (t , ω) = (z1 (t , ω1 ), z2 (t , ω2 )). Next we will prove the existence of a random attractor. First, we give the following two lemmas to prove the existence of a compact absorbing set. Lemma 5.1. Assume that α ∈ ( 21 , 1] and β ∈ (0, 1]. There exists two random radii r1 (ω) and r2 (ω) > 0 such that, for any given R > 0, there exists s1 (ω) ≤ −1 such that for all s ≤ s1 (ω), φs ∈ H × H satisfying ∥φs ∥W ≤ R, the following inequalities hold
∥φ ′ ∥2 ≤ r1 (ω),
∀t ∈ [−1, 0],
and
0
2R2 −1
∥φ ′ (τ )∥2V˙ dτ +
σ2 2
0
−1
z1−2 (τ )∥u′ (τ )∥4L4 dτ ≤ r22 (ω). ′
′
Proof. Multiplying the first term and the second term of Eq. (5.1) with u ∗ and v ∗ respectively, integrating over D and taking the real part lead to 1 d 2 dt
α ′ 2
∥u ∥ + γ2 ∥Λ u ∥ + σ2 z1 ′ 2
−2
1 1 1 2 ∥u ∥L4 ≤ γ1 + β1 ∥u′ ∥2 + z12 z2−2 ∥u′ ∥2 + ∥v ′ ∥2 , ′ 4
2
2
2
(5.4)
and 1 d 2 dt
β ′ 2
1
∥v ∥ + µ2 ∥Λ v ∥ ≤ −µ1 + β ′ 2
2
2 2
1
1
2
2
∥v ′ ∥2 + z1−2 z22 ∥u′ ∥2 + ∥v ′ ∥2 .
(5.5)
Combining (5.4)–(5.5) gives d dt
∥φ ′ ∥2W + 2γ2 ∥Λα u′ ∥2 + 2µ2 ∥Λβ v ′ ∥2 + 2σ2 z1−2
|u′ |4 dx
≤ (2γ1 + β12 )∥u′ ∥2 + (z12 z2−2 + z1−2 z22 )∥u′ ∥2 + (2 − 2µ1 + β22 )∥v ′ ∥2 . By Young’s inequality, we have
(3γ1 + β12 )∥u′ ∥2 ≤ σ2 z1−2 ∥u′ ∥4 +
π z16 z2−4 , 4 σ2 −2 4 z z π σ2 −2 ′ 4 z1−2 z22 ∥u′ ∥2 ≤ z1 ∥u ∥L4 + 1 2 , 4 σ2
z12 z2−2 ∥u′ ∥2 ≤
σ2
(3α1 + β1 )2 2 π z1 , 4σ2
z1−2 ∥u′ ∥4L4 +
then d dt
∥φ ′ ∥2W + 2γ2 ∥Λα u′ ∥2 + 2µ2 ∥Λβ v ′ ∥2 +
≤
σ2 2
z1−2
|u′ |4 dx + γ1 ∥u′ ∥2 + (2µ1 − 2 − β22 )∥v ′ ∥2
(3α1 + β1 )2 2 π z16 z2−4 z −2 z 4 π π z1 + + 1 2 , C0 z12 + C1 (z16 z2−4 + z1−2 z24 ). 4σ2 σ2 σ2
Let R8 = min{γ1 , 2µ1 − 2 − β22 } > 0. Then we have d dt
∥φ ′ ∥2W + R8 ∥φ ′ ∥2W ≤ C0 z12 + C1 (z16 z2−4 + z1−2 z24 ).
For any t ≥ s, it follows from the Gronwall lemma that
t t t ∥φ ′ ∥2W ≤ e−R8 t eR8 s ∥φ ′ (s)∥2W + C0 eR8 τ z12 (τ )dτ + C1 eR8 τ z16 (τ )z2−4 (τ )dτ + C1 eR8 τ z1−2 (τ )z24 (τ )dτ s s s t ≤ e−R8 t eR8 s (z12 (s) + z22 (s))∥φ(s)∥2W + C0 eR8 τ z12 (τ )dτ s t t + C1 eR8 τ z16 (τ )z2−4 (τ )dτ + C1 eR8 τ z1−2 (τ )z24 (τ )dτ . s
s
(5.6)
44
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
Thanks to W (s)
lim
s
s→−∞
= 0,
P-a.s,
therefore, eR8 s z12 (s) = eR8 s−2β1 W1 (s) → 0, eR8 s z22
R8 s−2β2 W2 (s)
(s) = e
P-a.s. as s → −∞,
→ 0,
P-a.s. as s → −∞.
Then for any φs ∈ H × H with ∥φs ∥W ≤ R, there exists a time s1 (ω) ≤ −1 such that eR8 s (z12 (s) + z22 (s))∥φs ∥2W ≤ 1,
P-a.s. ∀s ≤ s1 (ω).
We also have
W (s) < ε, as s(ω) ≤ s′1 (ω), we have s
∀ ε > 0, ∃ s′1 (ω) ≤ −1,
then e2β1 W1 (s) < e−2β1 εs , e
4β2 W (s)
−4β2 ε s
e−2β1 W1 (s) < e−2β1 εs ,
,
e
−4β2 W (s)
−4β2 ε s
e−6β1 W1 (s) < e−6β1 εs ,
.
Direct computation implies that t
C0
eR8 τ z12 (s)dτ ≤ C0
eR8 τ e−2β1 W1 (s) dτ ≤ C0
t
C1
eR8 τ e−2β1 ετ dτ ,
−∞ 0
eR8 τ z16 (τ )z2−4 (τ )dτ ≤ C1
eR8 τ e−(6β1 +4β2 )ετ dτ ,
−∞
s t
0
−∞
s
0
C1
0
eR8 τ z1−2 (τ )z24 (τ )dτ ≤ C1
eR8 τ e−(2β1 +4β2 )ετ dτ .
−∞
s
Let ε be small enough such that ε < 2β +84β and s1 (ω) = min{s1 (ω), s′1 (ω)}, hence it follows that 1 2 R
∥φ (t )∥ ′
2 W
≤e
−R 8 t
0
1 + C0
e
R8 τ −2β1 ετ
e
dτ + C1
−∞
≤ r12 (ω),
0
e
R8 τ −(6β1 +4β2 )ετ
e
dτ + C1
−∞
0
e
R8 τ −(2β1 +4β2 )ετ
e
dτ
−∞
t ∈ [−1, 0],
and integrating (5.6) from −1 to 0 gives 0
2R2
−1
∥φ ′ (τ )∥2V˙ dτ +
σ2 2
0
z12 (τ )dτ + C1
≤ C0 −1
0
−1
z1−2 (τ )∥u′ (τ )∥4L4 dτ
0
z16 (τ )z2−4 (τ )dτ + C1 −1
0
z1−2 (τ )z24 (τ )dτ + r12 (ω) , r22 (ω).
(5.7)
−1
Lemma 5.2. Assume that α ∈ ( 12 , 1] and β ∈ (0, 1]. There exists a random radius r3 (ω) > 0 such that for any given R > 0, there exists s1 (ω) ≤ −1 such that for all s ≤ s1 (ω), φs ∈ H × H satisfying ∥φs ∥W ≤ R, the following inequality hold
∥φ∥2V˙ ≤ r32 (ω),
∀t ∈ [−1, 0].
Proof. Multiplying the first term of Eq. (5.1) with (−∆)α u∗ , then integrating over D and taking the real parts, it follows from [13] and the Hölder inequality that ′
d dt
∥Λα u′ ∥2 + 2γ2 ∥Λ2α u′ ∥2 ≤ (2γ1 + β12 )∥Λα u′ ∥2 + z1−2 Cσ δ∥Λ2α u′ ∥2 + Cσ C (δ)z1−2 ∥Λα u′ ∥2 + CC (δ)∥Λα u′ ∥2 ∥u′ ∥4L4 +
Let δ =
γ2 −2 z1 Cσ
d dt
z12 z2−2
γ2
∥v ′ ∥2 + γ2 ∥Λ2α u′ ∥2 .
, then
∥Λα u′ ∥2 ≤ (2γ1 + β12 + Cσ C (δ)z1−2 + CC (δ)∥u′ ∥4L4 )∥Λα u′ ∥2 +
z12 z2−2
γ2
∥v ′ ∥2 .
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
45
Hence, we have d dt
z1−2 z22
∥Λβ v ′ (t )∥2 ≤ (β22 − 2µ1 )∥Λβ v ′ (t )∥2 +
2µ2
∥u′ ∥2 .
Then, d dt
∥φ ′ ∥2V˙ ≤ g1 (t )∥φ ′ ∥2V˙ + g2 (t ),
(5.8)
where g1 = 2γ1 + β12 + Cσ C (δ)z1−2 + CC (δ)∥u′ ∥4L4 > 0 > β22 − 2µ1 , and
g2 = R9 ∥φ ∥ , ′ 2
R9 = max
z12 z2−2 z1−2 z22
,
γ2
.
2µ2
Inequality (5.8) yields that for any −1 ≤ τ ≤ t ≤ 0,
∥φ ′ ∥2V˙
t
2 τ g1 (τ1 )dτ1 e V˙
≤ ∥φ (τ )∥ ≤ ∥φ ′ (τ )∥2V˙ + ′
+e
t
τ g1 (τ1 )dτ1
t
g2 (τ2 )e−
τ
0
g2 (τ2 )dτ2
0
e
−1 g (τ1 )dτ1
τ τ
2 g (τ )dτ 1 1
dτ2
.
−1
Integrating τ over [−1, 0], we obtain
∥φ ′ ∥2V˙ ≤
0
−1
∥φ ′ (τ )∥2V˙ dτ +
0
g2 (τ )dτ
e
0
−1 g1 (τ )dτ
.
−1
0
0
0
It follows from (5.7) that all the terms −1 ∥φ ′ (τ )∥2V˙ dτ , −1 g1 (τ )dτ and −1 g2 (τ )dτ are all bounded as s → −∞. Therefore, denote r32 (ω) ,
0
−1
∥φ ′ (τ )∥2V˙ dτ +
0
g2 (τ )dτ
0
e
−1 g1 (τ )dτ
,
−1
there exists s ≤ s1 (ω) such that
∥φ ′ ∥2V˙ ≤ r32 (ω), specially,
∥φ ′ (0)∥2V˙ = ∥φ(0)∥2V˙ ≤ r32 (ω). Let K (ω) be the ball in H α × H β with radius r3 (ω). Lemmas 5.1–5.2 indicate that for any bounded set B in H × H, there exists an s1 (ω) such that for any s ≤ s1 (ω), S (0, s; ω)B ⊂ K (ω) holds P-a.e.
(5.9) α
β
This implies that K (ω) is an attracting set at time 0 due to H × H ↩→ H × H. Applying Theorem 2.7, we have the following results. Theorem 5.3. Assume that α ∈ ( 12 , 1] and β ∈ (0, 1]. The stochastic dynamical system associated with the CFGL equation (1.4) has a random attractor in H × H. Acknowledgment We would like to thank an anonymous referee for the valuable suggestions for improving the quality of this paper. References [1] C. Doering, J. Gibbon, C. Levermore, Weak and strong solutions of the complex Ginzburg–Landau equation, Physica D 71 (1994) 285–318. [2] D. Dai, Z. Li, Z. Liu, Exact homoclinic wave and soliton solutions for the 2D Ginzburg–Landau equation, Phys. Lett. A 372 (17) (2008) 3010–3014. [3] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation, I. Compactness method, Physica D 95 (1) (1996) 191–228. [4] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation, II. Contraction method, Comm. Math. Phys. 187 (1) (1997) 45–79.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Well-posedness and dynamics of stochastic fractional model for nonlinear optical fiber materials✩ Tianlong Shen, Jianhua Huang ∗ College of Science, National University of Defense Technology, Changsha 410073, PR China
article
info
Article history: Received 11 April 2014 Accepted 21 June 2014 Communicated by Enzo Mitidieri MSC: 37L55 60H15 Keywords: Coupled fractional Ginzburg–Landau equation Fractional Laplacian Commutation estimates Well-posedness Random attractor
abstract The current paper is devoted to the well-posedness and dynamics of the stochastic coupled fractional Ginzburg–Landau equation, which describes a class of nonlinear optical fiber materials with active and passive coupled cores. By the commutation estimates and Fourier– Galerkin approximation, the global existence of weak solutions and the uniqueness criterion are established. Moreover, the existence of a global attractor is shown. Finally, we consider the long-time behavior of the stochastic coupled fractional Ginzburg–Landau equation (SCFGL) with multiplicative noise, and prove the existence of a random attractor for the random dynamical system generated by the SCFGL equation. © 2014 Elsevier Ltd. All rights reserved.
1. Fractional model for nonlinear optical fiber materials Ginzburg–Landau (GL) equations are usually applied to describe a class of optical fiber materials. There has been extensive study of the GL equations (see [1–6] and reference therein). The exact homoclinic wave and soliton solution of the GL equations have been studied in [2]. Guo et al. in [5] proved the existence of a global attractor for the GL equation. The coupled Ginzburg–Landau (CGL) equations have attracted considerable attention in modeling a class of nonlinear optical fiber materials with active and passive coupled cores. There are also many papers concerning the CGL equations (see [7–9] and reference therein). The existence of the stable solutions and exponential attractors for the CGL systems has been proved in [7,9] respectively. The fractional Laplacian operator is exactly the infinitesimal generators of Lévy stable diffusion processes, and there are many fractional models that arise in plasma, flames propagation and chemical reactions in liquids, geophysical fluid dynamics and financial market et al. There have been extensive study and application of fractional differential equations including the fractional Schrödinger equation [10], fractional Landau–Lifschitz equation [11], fractional Landau–Lifschitz–Maxwell equation [12] and fractional Ginzburg–Landau (FGL) equation [13]. Pu and Guo in [13] proved the well-posedness and dynamics for the FGL equation. However, there are some limitations to this model to describe some models with some perturbations which will lead to a very large complex system. In mathematical physics, the models can be described by stochastic partial differential equations. Based on [13], the dynamics for the stochastic FGL equation with multiplicative noise has been studied in [14]. ✩ This work was partially supported by NSFC (11371367, 11101427) and fundamental research project of NUDT (JC12-02-03).
∗
Corresponding author. Tel.: +86 73184574234. E-mail addresses: [email protected] (T. Shen), [email protected] (J. Huang).
http://dx.doi.org/10.1016/j.na.2014.06.018 0362-546X/© 2014 Elsevier Ltd. All rights reserved.
34
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
Motivated by [13,15], in the present paper, we consider the following coupled fractional Ginzburg–Landau (CFGL) equation: ut = γ1 u − (γ2 + iγ3 )(−∆)α u + (iσ1 − σ2 )|u|2 u + iv, x ∈ R, t > 0, vt = (−µ1 + iµ2 )v − (µ3 + iµ4 )(−∆)β v + iu, x ∈ R, t > 0,
(1.1)
with the initial conditions and the periodic boundary conditions: u(x, 0) = u0 (x), v(x, 0) = v0 (x), x ∈ R, u(x + 2π , t ) = u(x, t ), v(x + 2π , t ) = v(x, t ),
(1.2)
t > 0 , x ∈ R,
where α, β ∈ (0, 1), u and v denote the amplitude of the electromagnetic wave in a dual-core system, t denotes the time, x is the horizontal axis of the plane wave, γ2 > 0, µ2 > 0, µ3 > 0 and µ4 > 0 are dissipation coefficients and µ1 > 1, γ1 , γ3 , σ1 , σ2 are real numbers. The fractional Laplacian (−∆)α can be regarded as a pseudo differential operator with |ξ |2α and can be realized through the Fourier transform [16]:
u(ξ ), (− ∆)α u(ξ ) = |ξ |2α
(1.3) 1 2
where u is the Fourier transform of u. In what follows, we write Λ for (−∆) . However, there is a natural question: How about the dynamics for the stochastic CFGL equations? Motivated by [14], we also consider the following stochastic CFGL equation with multiplicative noise: du = [γ1 u − (γ2 + iγ3 )(−∆)α u + (iσ1 − σ2 )|u|2 u + iv]dt + β1 udW1 (t ), x ∈ R, t > 0, dv = [(−µ1 + iµ2 )v − (µ3 + iµ4 )(−∆)β v + iu]dt + β2 udW2 (t ), x ∈ R, t > 0,
(1.4)
with the same initial conditions and periodic boundary conditions to (1.1), where α, β ∈ (0, 1), γ1 > 0, γ2 > 0, β1 > 0,
µ1 > 1 +
β22
, µ3 > 0, β2 > 0, γ3 , σ1 , σ2 , µ2 and µ4 are real numbers and W1 (t ) and W2 (t ) are two-sided Wiener processes 2 on a complete probability space. The rest of the paper is organized as follows. The working function space and some basic concepts related to the random dynamical system are introduced in Section 2. We establish the well-posedness of the weak solutions for the deterministic CFGL equation in Section 3. In Section 4, we consider the long-time behavior of the solution and the existence of a global attractor is shown. Finally, a continuous random dynamical system for the stochastic CFGL equation is constructed and the existence of a random attractor is proved in Section 5. 2. Notations and preliminaries In this section, we first review some notations for the working function space. Denote H = L2per (D ) = {u|u ∈ L2 [0, 2π ], u(x + 2π , t ) = u(x, t )},
D = [0, 2π ],
W = H × H = {(u, v)|u ∈ H , v ∈ H }, with the norm
∥u∥2H = ⟨u, u∗ ⟩ =
|u|2 dx,
∥φ∥2W = ∥u∥2H + ∥v∥2H ,
D
where φ = (u, v) ∈ W . For simplicity, we use the notation ∥ · ∥ to represent the norm for space H. In what follows, we redefine some notations to the fractional derivative and fractional Sobolev space. Since u is a periodic function, it can be expressed by a Fourier series u(x) = (F −1 u)(x) :=
1 2π ξ ∈Z
u(ξ )eiξ x ,
where
u(ξ ) :=
e−iξ y u(y)dy. D
Then for s ∈ R, denote
Λs u = F −1 (|ξ |s u(ξ )).
˙ s under the norm Finally, for any s ∈ R, we define the homogeneous Sobolev space H ∥u∥H˙ s = ∥Λ u∥ = s
ξ ∈Z
|ξ | | u(ξ )| 2s
2
21
.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
35
Similarly, for any s ∈ R, we denote the inhomogeneous Sobolev space H s under the norm
∥u∥H s = ∥J s u∥ =
12 (1 + |ξ |2 )s | u(ξ )|2 . ξ ∈Z
Denote V = H α × H β = {(u, v)|u ∈ H α , v ∈ H β }, with the norm
∥φ∥2V = ∥u∥2H α + ∥v∥2H β , for any (u, v) ∈ V . Here are two lemmas about commutator estimates developed by Kato and Ponce in [17] and Kening et al. in [18], which are key technique tools to show the well-posedness of the weak solutions and dynamics for the CFGL equation. Lemma 2.1 ([17]). Assume that s > 0 and p ∈ (1, +∞). If f and g are the Schwartz class, then
∥Λs (fg ) − f Λs g ∥Lp ≤ C (∥∇ f ∥Lp1 ∥g ∥H˙ s−1,p2 + ∥f ∥H˙ s,p3 ∥g ∥Lp4 ), and
∥Λs (fg )∥Lp ≤ C (∥f ∥Lp1 ∥g ∥H˙ s,p2 + ∥f ∥H˙ s,p3 ∥g ∥Lp4 ), where p2 , p3 ∈ (1, +∞) such that 1 p
=
1 p1
+
1 p2
=
1 p3
+
1 p4
.
Lemma 2.2 ([18]). Assume that q > 1, p ∈ [q, +∞), x ∈ Rd and 1 p
+
s d
=
1 q
.
Suppose that Λs f ∈ Lq , then f ∈ Lp and there is a constant C ≥ 0 such that
∥f ∥Lp ≤ C ∥Λs f ∥Lq and if f = Λ−s g for g ∈ Lq , then
∥Λ−s g ∥Lp ≤ C ∥g ∥Lq . Finally, we recall some definitions related to the random dynamical system and random attractors. We can refer to [19] for more details. Definition 2.3. Let (Ω , F , P ) be a probability space. A measurable flow θ = {θt }t ∈R on Ω is defined as a mapping
θ : R × Ω → Ω, which is (B (R) × F ; F )-measurable and satisfies the flow property
θt θτ = θt +τ
for t , τ ∈ R,
and θ0 = idΩ . In addition, we suppose that the measure P is invariant with respect to the flow θ . Then the quadruple (Ω , F , P , θ ) is called a metric dynamical system (MDS). Definition 2.4. Let E be a complete and separable metric space. A random dynamical system (RDS) with space E carried by a metric dynamical system (Ω , F , P , θ ) is given by the mapping
ϕ : R+ × Ω × E → E , which is (B (R+ ) × F × B (E ); B (E ))-measurable and possesses the cocycle property:
ϕ(t + τ , ω, x) = ϕ(τ , θt ω, ϕ(t , ω, x)) ∀ t , τ ∈ R+ , x ∈ E , ω ∈ Ω , ϕ(0, ω, ·) = idΩ . Definition 2.5. (i) A set-valued mapping K : Ω → 2E taking value in the closed subsets of E is said to be measurable if for each x ∈ E the mapping ω → d(x, K (ω)) is measurable, where d(A, B) = sup inf d(x, y). x∈A y∈B
A measurable set-valued mapping K is called a random set.
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T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
(ii) Let A, B be random sets. A is said to attract B if d(ϕ(t , θ−t ω)B(θ−t ω), A(ω)) → 0,
as t → ∞ P-a.s.
A is said to absorb B if P-a.s. there exists an absorption time tB (ω) such that for all t ≥ tB (ω)
ϕ(t , θ−t ω)B(θ−t ω) ⊂ A(ω). (iii) The Ω -limit set of a random set K is defined by
ΩK (ω) =
ϕ(t , θ−t ω)K (θ−t ω).
T ≥0 t ≥T
Definition 2.6. A random attractor for an RDS ϕ is a compact random set A P-a.s. satisfying: (i) A is invariant, i.e., ϕ(t , ω)A(ω) = A(θt ω) for all t > 0; (ii) A attracts all the deterministic bounded sets in E. The following theorem (cf. [19, Theorem 3.11]) yields a sufficient criterion for the existence of a random attractor. Theorem 2.7. Let ϕ be an RDS and assume that there exists a compact random set K absorbing every deterministic bounded set B ⊂ E. Then the set A(ω) =
ΩB (ω),
(2.1)
B⊂E
is a random attractor for ϕ . 3. Well-posedness of the deterministic CFGL model In this section, we will prove the existence and uniqueness criteria of the weak solutions for the periodical CFGL equation (1.1) with initial data φ|t =0 = φ0 ∈ W . Theorem 3.1. For any φ0 ∈ W , there exists a weak solution for the CFGL equation (1.1) such that
φ ∈ C ([0, T ]; W ) ∩ L2 ([0, T ]; V ), and u ∈ L4 ([0, T ]; L4 (D )). To prove the theorem, we need several lemmas. Lemma 3.2. Let φ be a solution to the deterministic CFGL equation (1.1) with initial data φ0 ∈ W , then there hold
∥φ(t )∥2 ≤ e2R1 t ∥φ0 ∥2 ,
(3.1)
and
∥φ(t )∥ + 2R2 2
t
∥φ∥ τ + 2σ2 0
2 d V˙
t
0
∥u∥4L4 dτ ≤ e2R1 t ∥φ0 ∥2 ,
(3.2)
where R1 = max{γ1 + 1, 1 − µ1 } and R2 = min{γ2 , µ3 }. Proof. Multiplying the first equation and second one of systems (1.1) with u∗ and v ∗ respectively, integrating over D and taking the real part give 1 d 2 dt
∥u∥2 = γ1 ∥u∥2 − γ2 ∥Λα u∥2 − σ2 ∥u∥4L4 + Re 1
Ω
iv u∗ dx
≤ γ1 ∥u∥2 − γ2 ∥Λα u∥2 − σ2 ∥u∥4L4 + (∥u∥2 + ∥v∥2 ), 2
1 d 2 dt
1
∥v∥2 ≤ −µ1 ∥v∥2 − µ3 ∥Λβ v∥2 + (∥u∥2 + ∥v∥2 ). 2
(3.3) (3.4)
Combining (3.3) and (3.4) leads to d dt
∥φ∥2 + 2R2 ∥φ∥2V˙ + 2σ2 ∥u∥4L4 ≤ 2R1 ∥φ∥2 ,
where R1 = max{γ1 + 1, 1 − µ1 } and R2 = min{γ2 , µ3 }.
(3.5)
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
37
Getting rid of the positive terms on the left, by the Gronwall inequality we obtain
∥φ(t )∥2 ≤ e2R1 t ∥φ0 ∥2 .
(3.6)
Then integrating (3.5) over [0, t ] yields the estimate (3.2).
Lemma 3.3. Let φ be a solution for the CFGL equation (1.1), then we have
dφ dt 2
L ([0,t ],H −α ×H −β )
≤ C.
Proof. Multiplying the first equation of the system (1.1) with the test function ψ1∗ and then integrating over D ×[0, t ] lead to
t
du dt
0
t t t t ⟨iv, ψ1∗ ⟩. (iσ1 − σ2 )⟨|u|2 u, ψ1∗ ⟩ + (γ2 + iγ3 )⟨Λ2α u, ψ1∗ ⟩ + ⟨γ1 u, ψ1∗ ⟩ − , ψ1∗ = 0
0
0
0
By the Hölder inequality, we have
t ⟨γ1 u, ψ ∗ ⟩ ≤ C ∥u∥L2 ([0,t ];H ) ∥ψ1 ∥L4 ([0,t ];H α ) , 1 0 t (γ2 + iγ3 )⟨Λ2α u, ψ ∗ ⟩ ≤ C ∥u∥L2 ([0,t ];H α ) ∥ψ1 ∥L4 ([0,t ];H α ) , 1 0 t (iσ1 − σ2 )⟨|u|2 u, ψ ∗ ⟩ ≤ C ∥u∥34 ∥ψ1 ∥L4 ([0,T ];H α ) , 1 L ([0,t ];H )
(3.7)
(3.8)
(3.9)
0
t ⟨iv, ψ ∗ ⟩ ≤ C ∥v∥L2 ([0,t ];H ) ∥ψ1 ∥L4 ([0,t ];H α ) . 1
(3.10)
0
Combining (3.7)–(3.10) gives
t du ∗ , ψ 1 ≤ C ∥ψ1 ∥L4 ([0,T ];H α ) , dt
∀ψ1 ∈ L4 ([0, t ]; H α ).
(3.11)
∀ψ2 ∈ L4 ([0, t ]; H β ).
(3.12)
0
Similarly, we can obtain
t dv ∗ , ψ2 ≤ C ∥ψ2 ∥L4 ([0,t ];H β ) , dt 0
Combining (3.11) and (3.12) gives
t dφ ≤ C ∥ψ∥L4 ([0,t ];H α ×H β ) , , ψ dt
(3.13)
0
where ψ = (ψ1 , ψ2 ). Therefore,
dφ dt 2
L ([0,t ],H −α ×H −β )
dφ ≤ dt 4
L ([0,t ],H −α ×H −β )
Thus, the proof of Lemma 3.3 is completed.
≤ C.
Next it remains to show that ∥φ(t )∥W is a continuous function for t ≥ 0. Let Iψ (t ) = (⟨u(t ), ψ1∗ (t )⟩, ⟨v(t ), ψ2∗ (t )⟩). Lemma 3.4. Iψ (t ) is a continuous function of t for any ψ ∈ H × H. Proof. Multiplying the first equation and the second one of the CFGL equation (1.1) with ψ1∗ and ψ2∗ respectively, then integrating over D × [0, T ] give Iψ1 (t ) = ⟨u(t ), ψ1 ⟩ = ⟨u0 , ψ1 ⟩ + γ1
t
⟨u, ψ1 ⟩dτ − (γ2 + iγ3 ) 0 t 2 ∗ − (iσ1 − σ2 ) ⟨|u| u, ψ1 ⟩dτ + i ⟨v, ψ1∗ ⟩dτ , ∗
∗
∗
t
⟨Λ2α u, ψ1∗ ⟩dτ
0
0
Iψ2 (t ) = ⟨v(t ), ψ2∗ ⟩
= ⟨v0 , ψ2∗ ⟩ + (−µ1 + iµ2 )
0
t
⟨v, ψ2∗ ⟩dτ − (µ3 + iµ4 )
t
0
⟨Λ2β v, ψ2∗ ⟩dτ + i
t
⟨u, ψ2∗ ⟩dτ . 0
38
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
First, let ψ = (ψ1∗ , ψ2∗ ) ∈ C ∞ (D ) × C ∞ (D ) and 0 ≤ t1 < t2 ≤ T . According to [13], we have 1
|Iψ1 (t2 ) − Iψ1 (t1 )| ≤ (|γ1 | ∥ψ1 ∥L∞ + |γ2 + iγ3 | ∥Λ2α ψ1∗ ∥L∞ )∥u∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 + |iσ1 − σ2 | ∥ψ1 ∥L∞ ∥u∥3L4 ([t
1
4 1 ,t2 ];L )
1
|t1 − t2 | 4 + |i| ∥ψ1 ∥L∞ ∥v∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 ,
(3.14)
and 1
|Iψ2 (t2 ) − Iψ2 (t1 )| ≤ (| − µ1 + iµ2 | · ∥ψ2 ∥L∞ + |µ3 + iµ4 | · ∥Λ2β ψ2 ∥L∞ )∥v∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 1
+ ∥ψ2 ∥L∞ ∥u∥L2 ([t1 ,t2 ];H ) |t1 − t2 | 2 .
(3.15)
Combining (3.14)–(3.15) gives
|Iψ (t2 ) − Iψ (t1 )| ≤ |Iψ1 (t2 ) − Iψ1 (t1 )| + |Iψ2 (t2 ) − Iψ2 (t1 )| √ 1 ≤ Cψ 2|φ|L2 ([t1 ,t2 ];W ) |t1 − t2 | 2 + Cψ ∥u∥3L4 ([t √
1
3
1
4 1 ,t2 ];L ) 3
|t1 − t2 | 4 1
2 2Cψ eR1 T ∥φ0 ∥2W |t1 − t2 | 2 + Cψ e 2 R1 T ∥φ0 ∥W |t1 − t2 | 4 .
≤
(3.16)
Therefore, the continuity of Iψ (t ) follows. Next we use a density argument to extend to the general case for ψ ∈ H × H. Let ϵ > 0 be an arbitrary positive number, and for ψ ∈ H × H, we may select some ψϵ ∈ C ∞ (D ) × C ∞ (D ) such that ∥ψϵ − ψ∥W ≤ ϵ . By the triangle inequality and the Hölder inequality, we have
|Iψ (t2 ) − Iψ (t1 )| ≤ |⟨φ(t2 ), ψ⟩ − ⟨φ(t2 ), ψϵ ⟩| + |⟨φ(t1 ), ψϵ ⟩ − ⟨φ(t1 ), ψ⟩| + |⟨φ(t2 ), ψϵ ⟩ − ⟨φ(t1 ), ψϵ ⟩| ≤ ϵ(∥φ(t1 )∥W + ∥φ(t2 )∥W ) + |Iψϵ (t2 ) − Iψϵ (t1 )|.
(3.17)
Since Iψϵ is continuous in t , ∥φ(t1 )∥W and ∥φ(t2 )∥W are independent of ϵ and ϵ is arbitrary, the continuity of Iψ (t ) follows for any ψ ∈ H × H. We are now in a position to prove Theorem 3.1. Proof. The existence of the weak solutions can be proved by Fourier–Galerkin approximation and compactness argument. Let {e1 , e2 , . . . , eN } be an orthogonal basis of H and PN be the orthogonal projection onto its subspace span {e1 , e2 , . . . , eN }. We construct the two abstract ODEs in this finite-dimensional space that approximate the CFGL equation
t ⟨uN , ψ1∗ ⟩dτ − (γ2 + iγ3 ) ⟨Λ2α uN , ψ1∗ ⟩dτ 0 0 t 2 ∗ − (iσ1 − σ2 ) ⟨|uN | uN , ψ1 ⟩dτ + i ⟨vN , ψ1∗ ⟩dτ ,
⟨uN (t ), ψ1∗ ⟩ − ⟨u0N , ψ1∗ ⟩ = γ1
t
(3.18)
0
t t ⟨vN (t ), ψ2∗ ⟩ − ⟨v0N , ψ2∗ ⟩ = (−µ1 + iµ2 ) ⟨vN , ψ2∗ ⟩dτ − (µ3 + iµ4 ) ⟨Λ2β vN , ψ2∗ ⟩dτ 0 0 t ∗ + i ⟨uN , ψ2 ⟩dτ ,
(3.19)
0
where ψ ∈ C ∞ (D ) × C ∞ (D ). For any fixed T > 0, Lemmas 3.2–3.4 also hold for φN , in particular
∥φN (t )∥ + 2γ2 2
t
α
∥Λ uN ∥ dτ + 2µ3 2
0
≤ ∥φ0N ∥2 + 2R1
0
t
β
∥Λ vN ∥ dτ + 2σ2 2
t
0
∥uN ∥4L4 dτ
t
∥φN ∥2 dτ .
(3.20)
0
It follows that {φN } is bounded in L2 (0, T ; V ) and {uN } is bounded in L4 (0, T ; L4 ), from which we know {φN } is weakly compact and there exists a subsequence (still denoted as {φN }) such that φN → φ weakly in L2 (0, T ; V ), also {uN } → uN weakly dφ in L4 (0, T ; L4 ). In addition, Lemmas 3.2–3.4 show {φN } is weakly compact in W for every t ≥ 0 and dtn is bounded in L2 (0, T ; −α −β 2 H × H ). Therefore, {φN } is compact in L (0, T ; W ) (thanks to Theorem 5.1 in [20]). We obtain that ⟨φN , ψ ∗ ⟩ is a continuous function for t ≥ 0 from Lemma 3.4. Since the estimates (3.16) and (3.17) are independent of N, it is equicontinuous in C ([0, T ]) for every ψ ∈ W . On the other hand, it follows from Lemma 3.2 that {⟨φN , ψ ∗ ⟩} is uniformly bounded in C [0, T ]. Therefore, {⟨φN , ψ ∗ ⟩} is compact in C [0, T ]. From the above argument, we know that ⟨φN , ψ ∗ ⟩ is a continuous function for t ≥ 0 and it converges to ⟨φ, ψ ∗ ⟩ for every t ≥ 0. From [13] we know that |u2N |uN converges weakly to |u2 |u in L1 (0, T ; L1 ) and the convergence for the linear is obvious. Therefore, the limit φ satisfies the CFGL equation (1.1) in the sense of (3.18)–(3.19).
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
39
We end this section with the proof of the uniqueness criteria of the weak solutions. Theorem 3.5. Assume that α ∈ ( 12 , 1], β ∈ (0, 1) and T > 0, there is at most one solution for the deterministic CFGL equation (1.1) such that
φ ∈ C (0, T ; W ) ∩ L2 (0, T ; V ), if 2
u ∈ L 1−θ (0, T ; L4 ),
θ=
1 4α
.
Proof. Let uA and uB be two solutions for the first term of Eq. (1.1) and vA and vB be two solutions for the second term of Eq. (1.1), then w1 = uA − uB and w2 = vA − vB satisfy
w1t = γ1 w1 − (γ2 + iγ3 )(−∆)α w1 + (iσ1 − σ2 )|u|2 u + iw2 , w2t = (−µ1 + iµ2 )w2 − (µ3 + iµ4 )(−∆)β w2 + iw1 . It follows from [13] that d dt
∥w1 ∥ ≤ C γ1 + ∥U ∥ + ∥U ∥ 2
2 L4
2 1−θ 4 L
∥w1 ∥2 + ∥w1 ∥2 + ∥w2 ∥2 ,
(3.21)
where ∥U ∥2 = ∥uA ∥2 + ∥uB ∥2 . It is easy to obtain d dt
∥w2 ∥2 ≤ −2µ1 ∥w2 ∥2 + ∥w1 ∥2 + ∥w2 ∥2 .
(3.22)
Combining (3.21)–(3.22) leads to d dt
∥ω∥
2 W
≤ C γ1 + ∥U ∥ + ∥U ∥ 2 L4
2 1−θ 4 L
+ 2 ∥w1 ∥2 + 2(1 − µ1 )∥w2 ∥2 ,
(3.23)
where w = (w1 , w2 ). Therefore, d dt
∥ω∥2W ≤ R3 ∥w∥2W ,
(3.24)
where
R3 = max C
2
γ1 + ∥U ∥2L4 + ∥U ∥L14−θ
+ 2, 2(1 − µ1 ) .
The uniqueness of the solution follows from the Gronwall inequality. Thus, the proof is completed.
4. Global attractors of the deterministic CFGL model In this section, we first establish that the solutions are globally smooth without restrictions on parameters. Theorem 4.1. Let α ∈ ( 12 , 1] and β ∈ (0, 1]. Then for all s ≥ 0, the solutions of Eq. (1.1) with initial data φ0 ∈ H s × H s obey
∥Λs φ(t )∥W ≤ C where C is a constant depending on T , ∥φ0 ∥H s ×H s and the parameters. Proof. Multiplying the first term of Eq. (1.1) with (−∆)s u∗ , then integrating over D and taking the real part, we obtain d dt
∥Λ u∥ + 2γ2 ∥Λ s
2
s+α
2 2s ∗ 2s ∗ u∥ = 2γ1 ∥Λ u∥ + 2Re (iσ1 − σ2 ) |u| uΛ u dx + 2Re i v Λ u dx . 2
s
2
According to [13], we can get the following result by the Hölder inequality and Lemmas 2.1–2.2.
2 |u|2 uΛ2s u∗ dx ≤ δ∥Λα+s ∥2 + C (δ)∥Λs u∥2 + C ∥Λs u∥ ∥u∥44 . L It follows from the Hölder inequality that
i v Λ2s u∗ dx ≤ 1 ∥Λs u∥2 + 1 ∥Λs v∥2 . 2 2
40
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
We derive that d dt
∥Λs u∥2 + 2γ2 ∥Λs+α u∥2 ≤ Cσ δ∥Λs+α u∥2 + (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 1)∥Λs u∥2 + ∥Λs v∥2 .
2 γ2 , Cσ
Let δ = d
dt
then
∥Λs u∥2 ≤ (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 1)∥Λs u∥2 + ∥Λs v∥2 .
(4.1)
Multiplying the second term of Eq. (1.1) with (−∆)s v ∗ , then integrating over D and taking the real part give d dt
∥Λs v∥2 + 2µ3 ∥Λs+β v∥2 ≤ −2µ1 ∥Λs v∥2 + ∥Λs v∥2 + ∥Λs u∥2 .
(4.2)
Combining (4.1)–(4.2) leads to d dt
∥Λs φ∥2W ≤ R4 ∥Λs φ∥2 ,
(4.3)
where R4 = max{2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 + 2, 2(1 − µ1 )}. Since ∥u(t )∥4L4 is integrable, applying the Gronwall inequality gives
∥Λs φ∥2W ≤ ∥Λs φ(0)∥2W e
t
0 R4 (s)ds
< ∞,
∀s ≥ 0, t ∈ [0, T ].
(4.4)
Next we consider the long-time behavior in H × H for the solutions of the CFGL equation (1.1). The basic concepts and properties of attractors can be found in [21]. We have the following theorem. Theorem 4.2. Assume that α ∈ ( 21 , 1] and β ∈ (0, 1]. Then the solution operator S : S (t )φ0 = φ(t ) for all t > 0 of the CFGL equation (1.1) well defines a semigroup in the space H × H and the following statements hold: (i) (ii) (iii) (iv)
For any t > 0, S (t ) is continuous in H × H; For any φ0 ∈ H × H, S is continuous from [0, T ] to H × H; For any t > 0, S (t ) is compact in H × H; The semigroup {S (t )}t ≥0 possesses a global attractor A in H × H.
Proof. The results in Section 3 show that the solution φ(t ) for the CFGL equation (1.1) well defines a semigroup S (t ) on H × H. First, we consider the absorbing set in H × H. By elementary estimates, we obtain 1 2
1
1
2
2
∥u∥2 + γ2 ∥Λα u∥2 + σ2 ∥u∥4L4 − γ1 ∥u∥2 − ∥u∥2 − ∥v∥2 ≤ 0,
and 1 2
1
1
2
2
∥v∥2 + µ3 ∥Λβ v∥2 + µ1 ∥v∥2 − ∥u∥2 − ∥v∥2 ≤ 0.
Thus, we obtain d dt
∥φ∥2W + 2γ2 ∥Λα u∥2 + 2µ3 ∥Λβ v∥2 − 2(γ1 + 1)∥u∥2 − 2(1 − µ1 )∥v∥2 + 2σ2 ∥u∥4L4 ≤ 0.
(Case i) If γ1 + 1 < 0, 1 − µ1 < 0, then d dt
∥φ∥2W − 2R5 ∥φ∥2W ≤ 0,
where R5 = max{γ1 + 1, 1 − µ1 } < 0. Therefore,
∥φ∥W ≤ ∥φ0 ∥W eR5 t , from which it follows that for any φ0 ∈ W ,
∥φ∥W → 0,
as t → ∞.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
41
(Case ii) If γ1 + 1 = 0, 1 − µ1 = 0, the direct result leads to
∥φ∥W ≤ ∥φ0 ∥W ,
as t → ∞.
(Case iii) If γ1 + 1 > 0, 1 − µ1 < 0, by Young’s inequality, we have 1 2
σ2 ∥u∥L4 +
4π
σ2
(γ1 + 1)2 ≥ 2(γ1 + 1)∥u∥2 ,
then d dt
∥φ∥2W + 2γ2 ∥Λα u∥2 + 2µ3 ∥Λβ v∥2 + 2(γ1 + 1)∥u∥2 + 2(µ1 − 1)∥v∥2 + σ2 ∥u∥4L4 ≤
8π (γ1 + 1)2
σ2
.
(4.5)
Therefore, d dt
8π (γ1 + 1)2
∥φ∥2W + R6 ∥φ∥2W ≤
σ2
,
where R6 = min{2(γ1 + 1), 2(µ1 − 1)} > 0. Applying the Gronwall inequality gives
∥φ∥2W ≤ e−R6 t ∥φ0 ∥2W +
8π (γ1 + 1)2
σ2
(1 − e−R6 t ) ∀t ≥ 0.
(4.6)
Therefore, lim ∥φ∥2W ≤ ρ02 ,
ρ02 =
t →∞
8π (γ1 + 1)2
σ2
.
We can infer the existence of an absorbing ball in H × H from (4.6). For any ρ > ρ0 , the balls B(0, ρ) are positively invariants for the semigroup S (t ) associated with the CFGL equation (1.1) and these balls are absorbing in H × H. Denote B0 the ball B(0, ρ0′ ) with any fixed ρ0′ > ρ0 . For any bounded set B , there exists a positive t0 = B(0, ρ0 ) for t ≥ t0 = t0 (B , ρ0 ). Integrating (4.5) from t to t + 1 leads to ′
′
∥φ(t + 1)∥2W + 2R2
t +1
∥φ∥2V˙ ds + σ2
t
8π (γ1 + 1)2
t +1
∥u∥4L4 ds ≤ ∥φ(t )∥2W +
t
σ2
1 R
log
ρ2 ρ0′2 −ρ02
such that S (t )B ⊂
.
Then it follows from (4.6) that t +1
∥φ∥2V˙ ds + σ2
2R2 t
t +1
′
∥u∥4L4 ds ≤ ρ02 +
t
8π (γ1 + 1)2
σ2
,
∀t ≥ t0 .
So for all t ≥ t0 , t +1
∥φ∥2V˙ ds ≤ a1 ,
t
t +1
t
∥u∥4L4 ds ≤ a2 ,
(4.7)
are uniformly bounded independent of φ0 . Next we consider the absorbing set in H α × H β . Similarly to the proof of Theorem 4.1, multiplying the first term and the second term of Eq. (1.1) with (−∆)α u∗ and (−∆)β v ∗ respectively, then integrating over D and taking the real parts, it follows from [13] and the Hölder inequality that d dt
∥Λα u∥2 + 2γ2 ∥Λ2α u∥2 ≤ (2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 )∥Λα u∥2 +
1
γ2
∥v∥2 + (Cσ δ + γ2 )∥Λ2α u∥2 ,
(4.8)
and d dt
∥Λβ v∥2 + 2µ3 ∥Λ2β v∥2 ≤ −2µ1 ∥Λβ v∥2 + 2µ3 ∥Λ2β v∥2 +
Let δ = d dt
γ2 Cσ
1 2µ3
∥ u∥ 2 .
(4.9)
. Then combining (4.8)–(4.9) gives
∥φ∥2V˙ ≤ h1 ∥φ∥2V˙ + h2 ,
(4.10)
42
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
where 2µ1 < 0 < h1 = 2γ1 + Cσ C (δ) + CCσ ∥u∥4L4 , and h2 = R7 ∥φ∥2W ,
R7 = max
1
,
1
2µ3 γ2
.
By (4.7), we obtain t +1
h1 (s)ds ≤ 2γ1 + Cσ C (δ) + CCσ a2 , a3 ,
t +1
′
h2 (s)ds ≤ R7 ρ02 , a4 ,
∀t ≥ t0 .
(4.11)
t
t
Applying the uniform Gronwall lemma [21] leads to d dt
∥φ∥2V˙ ≤ (a1 + a4 )ea3 ,
∀t ≥ t0 + 1.
(4.12)
The result (4.12) can assure us the existence of an absorbing ball of the solutions in H α × H β . Let φ0 ∈ B be a bounded set. Since B1 is bounded in H α × H β , and the embedding H α × H β ↩→ H × H is compact, we obtain
S (t )B is relative compact in H × H .
(4.13)
t ≥ t0 + 1
Thus, the assertion of item (iii) has been proved. Notice that items (i)–(ii) can be verified similar to one in Section 3, and item (iv) is a direct corollary of items (i)–(iii). Thus, the proof is completed. 5. Stochastic CFGL model with multiplicative noise In this section, we will establish the well-posedness and random dynamical system for the stochastic CFGL equation (1.4) with multiplicative noise of Itô form in R. Due to the special linear multiplicative noise, the stochastic CFGL equation (1.4) can be reduced to the equations with random coefficients by a suitable change of variable in the following form:
z1 (t ) = e−β1 W1 (t ) , z2 (t ) = e−β2 W2 (t ) ,
satisfy the stochastic differential equations:
1 dz1 (t ) = β12 z1 dt − β1 z1 (t )dW1 (t ), 2
dz2 (t ) = 1 β 2 z1 dt − β2 z2 (t )dW2 (t ). 2 2
We translate the unknown u′ (t ) = z1 (t )u(t ), v ′ (t ) = z2 (t )v(t ) to obtain the following random differential equations:
′ 1 2 du (t ) = γ + β u′ − (γ2 + iγ3 )(−∆)α u′ + (iσ1 − σ2 )z1−2 |u′ |2 u′ + iz1 z2−1 v ′ , 1 1 dt 2 dv ′ (t ) 1 = −µ1 + iµ2 + β22 v ′ − (µ3 + iµ4 )(−∆)β v ′ + iz1−1 z2 u′ , dt
(5.1)
2
with the initial conditions at time s and the periodic boundary conditions:
u′ (x, s) = us (x), v ′ (x, s) = vs (x), x ∈ R, u′ (x + 2π , t ) = u(x, t ), v ′ (x + 2π , t ) = v(x, t ),
t > 0, x ∈ R.
(5.2)
In what follows, we construct a random dynamical system modeling the stochastic CFGL equation (5.1). First, we consider the set of continuous functions with value 0 at 0,
Ω1 = {ω1 ∈ C (R, R) : ω1 (0) = 0},
Ω2 = {ω2 ∈ C (R, R) : ω2 (0) = 0}.
Let W1 (t , ω1 ) = ω1 (t ) and W2 (t , ω2 ) = ω2 (t ), (Ω1 , F , P) and (Ω2 , F , P) be complete probability space. A family of measure preserving and ergodic transformation of (Ω1 , F , P, (θt )t ∈R ) and (Ω2 , F , P, (θt )t ∈R ) can be defined by
θt ω1 (s) = ω1 (t + s) − ω1 (s),
θt ω2 (s) = ω2 (t + s) − ω2 (s).
For the Winner process Wj (t ), it follows that E [|Wj (t ) − Wj (s)|4 ] ≤ 3|t − s|,
j = 1, 2.
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
43
j , which is a continuous function, such that P ({W j (t , ω) ̸= Due to the Kolmogorov Theorem, there exists an equal process W Wj (t , ω)}) = 0. Thus, for any fixed ω = (ω1 , ω2 ) ∈ Ω1 × Ω2 , the existence and uniqueness of the solution for the stochastic CFGL equation (1.4) can be obtained by the same method in Section 3, which defines a stochastic dynamical system (S (t , s; ω))t ≥s,ω∈Ω1 ×Ω2 by S (t , s; ω)φs = φ(t , ω; s, φs ) = φ ′ (t , ω; s, φs z (s, ω))z (t , ω),
(5.3)
where z (t , ω) = (z1 (t , ω1 ), z2 (t , ω2 )). Next we will prove the existence of a random attractor. First, we give the following two lemmas to prove the existence of a compact absorbing set. Lemma 5.1. Assume that α ∈ ( 21 , 1] and β ∈ (0, 1]. There exists two random radii r1 (ω) and r2 (ω) > 0 such that, for any given R > 0, there exists s1 (ω) ≤ −1 such that for all s ≤ s1 (ω), φs ∈ H × H satisfying ∥φs ∥W ≤ R, the following inequalities hold
∥φ ′ ∥2 ≤ r1 (ω),
∀t ∈ [−1, 0],
and
0
2R2 −1
∥φ ′ (τ )∥2V˙ dτ +
σ2 2
0
−1
z1−2 (τ )∥u′ (τ )∥4L4 dτ ≤ r22 (ω). ′
′
Proof. Multiplying the first term and the second term of Eq. (5.1) with u ∗ and v ∗ respectively, integrating over D and taking the real part lead to 1 d 2 dt
α ′ 2
∥u ∥ + γ2 ∥Λ u ∥ + σ2 z1 ′ 2
−2
1 1 1 2 ∥u ∥L4 ≤ γ1 + β1 ∥u′ ∥2 + z12 z2−2 ∥u′ ∥2 + ∥v ′ ∥2 , ′ 4
2
2
2
(5.4)
and 1 d 2 dt
β ′ 2
1
∥v ∥ + µ2 ∥Λ v ∥ ≤ −µ1 + β ′ 2
2
2 2
1
1
2
2
∥v ′ ∥2 + z1−2 z22 ∥u′ ∥2 + ∥v ′ ∥2 .
(5.5)
Combining (5.4)–(5.5) gives d dt
∥φ ′ ∥2W + 2γ2 ∥Λα u′ ∥2 + 2µ2 ∥Λβ v ′ ∥2 + 2σ2 z1−2
|u′ |4 dx
≤ (2γ1 + β12 )∥u′ ∥2 + (z12 z2−2 + z1−2 z22 )∥u′ ∥2 + (2 − 2µ1 + β22 )∥v ′ ∥2 . By Young’s inequality, we have
(3γ1 + β12 )∥u′ ∥2 ≤ σ2 z1−2 ∥u′ ∥4 +
π z16 z2−4 , 4 σ2 −2 4 z z π σ2 −2 ′ 4 z1−2 z22 ∥u′ ∥2 ≤ z1 ∥u ∥L4 + 1 2 , 4 σ2
z12 z2−2 ∥u′ ∥2 ≤
σ2
(3α1 + β1 )2 2 π z1 , 4σ2
z1−2 ∥u′ ∥4L4 +
then d dt
∥φ ′ ∥2W + 2γ2 ∥Λα u′ ∥2 + 2µ2 ∥Λβ v ′ ∥2 +
≤
σ2 2
z1−2
|u′ |4 dx + γ1 ∥u′ ∥2 + (2µ1 − 2 − β22 )∥v ′ ∥2
(3α1 + β1 )2 2 π z16 z2−4 z −2 z 4 π π z1 + + 1 2 , C0 z12 + C1 (z16 z2−4 + z1−2 z24 ). 4σ2 σ2 σ2
Let R8 = min{γ1 , 2µ1 − 2 − β22 } > 0. Then we have d dt
∥φ ′ ∥2W + R8 ∥φ ′ ∥2W ≤ C0 z12 + C1 (z16 z2−4 + z1−2 z24 ).
For any t ≥ s, it follows from the Gronwall lemma that
t t t ∥φ ′ ∥2W ≤ e−R8 t eR8 s ∥φ ′ (s)∥2W + C0 eR8 τ z12 (τ )dτ + C1 eR8 τ z16 (τ )z2−4 (τ )dτ + C1 eR8 τ z1−2 (τ )z24 (τ )dτ s s s t ≤ e−R8 t eR8 s (z12 (s) + z22 (s))∥φ(s)∥2W + C0 eR8 τ z12 (τ )dτ s t t + C1 eR8 τ z16 (τ )z2−4 (τ )dτ + C1 eR8 τ z1−2 (τ )z24 (τ )dτ . s
s
(5.6)
44
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
Thanks to W (s)
lim
s
s→−∞
= 0,
P-a.s,
therefore, eR8 s z12 (s) = eR8 s−2β1 W1 (s) → 0, eR8 s z22
R8 s−2β2 W2 (s)
(s) = e
P-a.s. as s → −∞,
→ 0,
P-a.s. as s → −∞.
Then for any φs ∈ H × H with ∥φs ∥W ≤ R, there exists a time s1 (ω) ≤ −1 such that eR8 s (z12 (s) + z22 (s))∥φs ∥2W ≤ 1,
P-a.s. ∀s ≤ s1 (ω).
We also have
W (s) < ε, as s(ω) ≤ s′1 (ω), we have s
∀ ε > 0, ∃ s′1 (ω) ≤ −1,
then e2β1 W1 (s) < e−2β1 εs , e
4β2 W (s)
−4β2 ε s
e−2β1 W1 (s) < e−2β1 εs ,
,
e
−4β2 W (s)
−4β2 ε s
e−6β1 W1 (s) < e−6β1 εs ,
.
Direct computation implies that t
C0
eR8 τ z12 (s)dτ ≤ C0
eR8 τ e−2β1 W1 (s) dτ ≤ C0
t
C1
eR8 τ e−2β1 ετ dτ ,
−∞ 0
eR8 τ z16 (τ )z2−4 (τ )dτ ≤ C1
eR8 τ e−(6β1 +4β2 )ετ dτ ,
−∞
s t
0
−∞
s
0
C1
0
eR8 τ z1−2 (τ )z24 (τ )dτ ≤ C1
eR8 τ e−(2β1 +4β2 )ετ dτ .
−∞
s
Let ε be small enough such that ε < 2β +84β and s1 (ω) = min{s1 (ω), s′1 (ω)}, hence it follows that 1 2 R
∥φ (t )∥ ′
2 W
≤e
−R 8 t
0
1 + C0
e
R8 τ −2β1 ετ
e
dτ + C1
−∞
≤ r12 (ω),
0
e
R8 τ −(6β1 +4β2 )ετ
e
dτ + C1
−∞
0
e
R8 τ −(2β1 +4β2 )ετ
e
dτ
−∞
t ∈ [−1, 0],
and integrating (5.6) from −1 to 0 gives 0
2R2
−1
∥φ ′ (τ )∥2V˙ dτ +
σ2 2
0
z12 (τ )dτ + C1
≤ C0 −1
0
−1
z1−2 (τ )∥u′ (τ )∥4L4 dτ
0
z16 (τ )z2−4 (τ )dτ + C1 −1
0
z1−2 (τ )z24 (τ )dτ + r12 (ω) , r22 (ω).
(5.7)
−1
Lemma 5.2. Assume that α ∈ ( 12 , 1] and β ∈ (0, 1]. There exists a random radius r3 (ω) > 0 such that for any given R > 0, there exists s1 (ω) ≤ −1 such that for all s ≤ s1 (ω), φs ∈ H × H satisfying ∥φs ∥W ≤ R, the following inequality hold
∥φ∥2V˙ ≤ r32 (ω),
∀t ∈ [−1, 0].
Proof. Multiplying the first term of Eq. (5.1) with (−∆)α u∗ , then integrating over D and taking the real parts, it follows from [13] and the Hölder inequality that ′
d dt
∥Λα u′ ∥2 + 2γ2 ∥Λ2α u′ ∥2 ≤ (2γ1 + β12 )∥Λα u′ ∥2 + z1−2 Cσ δ∥Λ2α u′ ∥2 + Cσ C (δ)z1−2 ∥Λα u′ ∥2 + CC (δ)∥Λα u′ ∥2 ∥u′ ∥4L4 +
Let δ =
γ2 −2 z1 Cσ
d dt
z12 z2−2
γ2
∥v ′ ∥2 + γ2 ∥Λ2α u′ ∥2 .
, then
∥Λα u′ ∥2 ≤ (2γ1 + β12 + Cσ C (δ)z1−2 + CC (δ)∥u′ ∥4L4 )∥Λα u′ ∥2 +
z12 z2−2
γ2
∥v ′ ∥2 .
T. Shen, J. Huang / Nonlinear Analysis 110 (2014) 33–46
45
Hence, we have d dt
z1−2 z22
∥Λβ v ′ (t )∥2 ≤ (β22 − 2µ1 )∥Λβ v ′ (t )∥2 +
2µ2
∥u′ ∥2 .
Then, d dt
∥φ ′ ∥2V˙ ≤ g1 (t )∥φ ′ ∥2V˙ + g2 (t ),
(5.8)
where g1 = 2γ1 + β12 + Cσ C (δ)z1−2 + CC (δ)∥u′ ∥4L4 > 0 > β22 − 2µ1 , and
g2 = R9 ∥φ ∥ , ′ 2
R9 = max
z12 z2−2 z1−2 z22
,
γ2
.
2µ2
Inequality (5.8) yields that for any −1 ≤ τ ≤ t ≤ 0,
∥φ ′ ∥2V˙
t
2 τ g1 (τ1 )dτ1 e V˙
≤ ∥φ (τ )∥ ≤ ∥φ ′ (τ )∥2V˙ + ′
+e
t
τ g1 (τ1 )dτ1
t
g2 (τ2 )e−
τ
0
g2 (τ2 )dτ2
0
e
−1 g (τ1 )dτ1
τ τ
2 g (τ )dτ 1 1
dτ2
.
−1
Integrating τ over [−1, 0], we obtain
∥φ ′ ∥2V˙ ≤
0
−1
∥φ ′ (τ )∥2V˙ dτ +
0
g2 (τ )dτ
e
0
−1 g1 (τ )dτ
.
−1
0
0
0
It follows from (5.7) that all the terms −1 ∥φ ′ (τ )∥2V˙ dτ , −1 g1 (τ )dτ and −1 g2 (τ )dτ are all bounded as s → −∞. Therefore, denote r32 (ω) ,
0
−1
∥φ ′ (τ )∥2V˙ dτ +
0
g2 (τ )dτ
0
e
−1 g1 (τ )dτ
,
−1
there exists s ≤ s1 (ω) such that
∥φ ′ ∥2V˙ ≤ r32 (ω), specially,
∥φ ′ (0)∥2V˙ = ∥φ(0)∥2V˙ ≤ r32 (ω). Let K (ω) be the ball in H α × H β with radius r3 (ω). Lemmas 5.1–5.2 indicate that for any bounded set B in H × H, there exists an s1 (ω) such that for any s ≤ s1 (ω), S (0, s; ω)B ⊂ K (ω) holds P-a.e.
(5.9) α
β
This implies that K (ω) is an attracting set at time 0 due to H × H ↩→ H × H. Applying Theorem 2.7, we have the following results. Theorem 5.3. Assume that α ∈ ( 12 , 1] and β ∈ (0, 1]. The stochastic dynamical system associated with the CFGL equation (1.4) has a random attractor in H × H. Acknowledgment We would like to thank an anonymous referee for the valuable suggestions for improving the quality of this paper. References [1] C. Doering, J. Gibbon, C. Levermore, Weak and strong solutions of the complex Ginzburg–Landau equation, Physica D 71 (1994) 285–318. [2] D. Dai, Z. Li, Z. Liu, Exact homoclinic wave and soliton solutions for the 2D Ginzburg–Landau equation, Phys. Lett. A 372 (17) (2008) 3010–3014. [3] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation, I. Compactness method, Physica D 95 (1) (1996) 191–228. [4] J. Ginibre, G. Velo, The Cauchy problem in local spaces for the complex Ginzburg–Landau equation, II. Contraction method, Comm. Math. Phys. 187 (1) (1997) 45–79.
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