Citation: Chunting Ji, Hui Liu, Jie Xin. Random attractors of the stochastic extended Brusselator system with a multiplicative noise[J]. AIMS Mathematics, 2020, 5(4): 3584-3611. doi: 10.3934/math.2020233
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In 1968, Prigogine and Lefever [1] firstly proposed the original Brusselator equations, which is a great significance reaction-diffusion system. It derives from the simulation of autocatalytic chemical or biochemical reactions, the formation of biological and cellular patterns, such as the chlorite-iodide-malonic acid reaction [2]. Up to 1993, Pearson [3] and Lee [4] discovered a large number of self-replicating pattern formation consistent with cubic-autocatalytic reaction-diffusion systems by experimental methods and numerical simulation methods, and showed that the Brusselator equations or Gray-Scott system [5,6] exhibit abundant spatial patterns.
Due to its wide application in biology and chemistry such as morphogenesis and trimolecular autocatalytic reactions, etc. Many researchers [7,8,9,10,11,12] have deeply studied the Brusselator equations. Guo and Han [7] studied attractor and proved spatial chaos by using the technique of unstable manifolds for the Brusselator system in RN. You [8] obtained the existence of a global attractor and proved the Hausdorff dimension and the fractal dimension are finite for the Brusselator equations, and established global attractor of a coupled two-cell Brusselator model in [9]. Bie [10] showed that if nonlinear term f has sublinear growth then no stationary patterns occur, while if f has superlinear growth, stationary patterns may exist for a general two-cell Brusselator model. Recently, Parshed et al. [11] firstly studied global existence of classical solutions, via construction of an appropriate Lyapunov functional for a four compartment Brusselator type system. Then, they proved global existence of weak solutions and obtained the existence of a global attractor.
In this article, we study the existence of a random attractor for the following stochastic extended Brusselator system:
{du=[d1Δu+a−(b+k)u+u2v+D1(w−u)+Nφ]dt+ρu∘dW(t),(1.1)dv=[d2Δv+bu−u2v+D2(z−v)]dt+ρv∘dW(t),(1.2)dφ=[d3Δφ+ku−(λ+N)φ+D3(ψ−φ)]dt+ρφ∘dW(t),(1.3)dw=[d1Δw+a−(b+k)w+w2z+D1(u−w)+Nψ]dt+ρw∘dW(t),(1.4)dz=[d2Δz+bw−w2z+D2(v−z)]dt+ρz∘dW(t),(1.5)dψ=[d3Δψ+kw−(λ+N)ψ+D3(φ−ψ)]dt+ρψ∘dW(t),(1.6) |
on R×O with the initial conditions
u(t0,x)=u0(x),v(t0,x)=v0(x),φ(t0,x)=φ0(x), | (1.7) |
w(t0,x)=w0(x),z(t0,x)=z0(x),ψ(t0,x)=ψ0(x),x∈O, | (1.8) |
and boundary conditions
u(t,x)=v(t,x)=φ(t,x)=w(t,x)=z(t,x)=ψ(t,x)=0,t>t0,x∈∂O, | (1.9) |
where O is a bounded domain in Rn(n≤3). Let d1,d2,d3,D1,D2,D3,a,b,k,λ and N are nonnegative numbers. Here W(t)t∈R is a two-sided real-valued Wiener process defined on the probability space, and the symbol ∘ represents the Stratonovich's integration.
In our study of the existence of random attractors for stochastic dynamics, we use the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms, cf. (2.13)–(2.18). Based on this, we have to overcome the difficulties of coupling structure and make more complex estimates.
It's noticing that six coupling components with partial reversibility constitute the extended Brusselator system, which to some extend reflects the relevant network dynamics, see, e.g., [13,14]. If ρ=0, system (1.1)–(1.6) reduces to the extended Brusselator system without random terms, which has been established the global dissipative dynamics by You and Zhou in [15]. Furthermore, Tu and You [16] proved random attractor of stochastic Brusselator system with a multiplicative noise, this paper has included the results of [16] when w,z,φ,ψ=0.
The structure of this article is as follows. In section 2, we will introduce an important theorem about the existence of random attractors and some basic facts. In section 3, we will study pullback absorbing property of the stochastic extended Brusselator equations. In section 4, we apply the uniform Gronwall inequality to prove the pullback asymptotic compactness. Then, we will get the main results of this paper in section 5.
In this section, we recall an essential theorem for the existence of random attractors. Please note that here we will not introduce some basic concepts associated with random attractors and stochastic dynamical systems. The reader can refer to [16,17,18,19,20,21,22,23,24,25] for these knowledge.
Let (X,||⋅||X) indicates a real separable Banach space with Borel σ-algebra B(X), and (Ω,F,P) is a probability space. Assume that (Ω,F,P,{θt}t∈R) is a metric dynamical system(MDS), and f is a continuous RDS on X over MDS.
Definition 2.1. A random set in X is a set-valued function B(ω):Ω→2X whose graph {(ω,x):x∈B(ω)}⊂Ω×X is an element of the product σ-algebra F×B(X). If one has a random variable r(ω)≥0 such that ||B(ω)||:=supx∈B(ω)||x||≤r(ω) for every ω∈Ω, then B(ω) is a bounded random set in X. If the set B(ω) is compact (or precompact) in X for all ω∈Ω, then the random set B(ω) is called compact (or precompact). The bounded random set B is called tempered with respect to (θt)t∈R on the probability space (Ω,F,P) means that for all ω∈Ω and every positive constant ν,
limt→∞e−νt||B(θ−tω)||=0. |
Besides, we say that a random variable R:(Ω,F,P)→(0,∞) is tempered if for any ω∈Ω
limt→±∞1tlogR(θ−tω)=0. |
A collection D of random subsets of X is called inclusion-closed, if D={D(ω)}ω∈Ω∈D and ˆD={ˆD (ω)⊂D(ω):ω∈Ω} means that ˆD ∈D. In this way, we say that the collection D is a universe. Let D indicates the universe of all the tempered random sets in X. Noticing that D contains each bounded non-random sets.
Proposition 2.1. (see [17,18]) Let D be a collection of random subsets of a Banach space X and f is a continuous RDS on X over a MDS. Assume that there is a closed pullback absorbing set {K(ω)}ω∈Ω and f is pullback asymptotically compact associated with D, thus the RDS f has a unique random attractor A={A(ω)}ω∈Ω whose basin is D and
A(ω)=⋂τ≥0¯⋃t≥τf(t,θ−tω,K(θ−tω)). | (2.1) |
Define the product Hilbert spaces
H=[L2(O)]6,E=[H1(O)]6,Π=[H10(O)∩H2(O)]6. |
We denote the norm and the inner product by ||⋅|| and ⟨⋅,⋅⟩, respectively. ||⋅||Lp(p≠2) represent the norm of Lp(O) or the product space Lp(O)=[Lp(O)]6. According to the Poincˊare inequality and the homogeneous Dirichlet boundary condition (1.9), one has a nonnegative constant γ such that
||∇ζ||2≥γ||ζ||2,∀ζ∈H10(O)orE, | (2.2) |
where ||∇ζ|| denotes the equivalent norm of the space E or the space H10(O).
The linear sectorial operator
A=(d1Δ000000d2Δ000000d3Δ000000d1Δ000000d2Δ000000d3Δ):Π→H | (2.3) |
denotes the generator of an analytic C0-semigroup on the Hilbert space H, see, e.g., [26].
Applying the Sobolev embedding theorem, H10(O)↪L6(O) is a continuous embedding for n≤3, therefore one has a nonnegative constant δ satisfies the following Sobolev imbedding inequality
||φ||L6(O)≤δ||φ||E=δ||∇φ||,∀φ∈E. | (2.4) |
Let
H(u,v,φ,w,z,ϕ)=(a−(b+k)u+u2v+D1(w−u)+Nφbu−u2v+D2(z−v)ku−(λ+N)φ+D3(ψ−φ)a−(b+k)w+w2z+D1(u−w)+Nψbw−w2z+D2(v−z)kw−(λ+N)ψ+D3(φ−ψ)):E→H | (2.5) |
is a locally Lipschitz continuous mapping on E. Then, the system (1.1)–(1.9) can be expressed as
dfdt=Af+H(f)+ρf∘dW(t)dt, | (2.6) |
f(t0,x)=f0(x)=(u0(x),v0(x),φ0(x),w0(x),z0(x),ϕ0(x)). | (2.7) |
Suppose that W(t)t∈R is a one-dimensional two-sided real-valued Wiener process defined on the probability space (Ω,F,P), where
Ω={ω∈C(R,R):ω(0)=0}, |
the Borel σ-algebra F on Ω is produced by the compact-open topology, and P is consistent with Wiener measure. The time shift defined on the probability space (Ω,F,P) is given by
θtω(⋅)=ω(t+⋅)−ω(t),t∈R. |
Therefore, (Ω,F,P,{θt}t∈R) constitute a metric dynamical system f, see Arnold [19].
In the rest of this section, we consider the following Ornstein-Uhlenbeck process
z(θtω)=−∫0−∞es(θtω)(s)ds=−∫0−∞esω(t+s)ds+ω(t), | (2.8) |
and z(θtω) satisfies the following linear stochastic differential equation
dz+zdt=dW(t). | (2.9) |
Proposition 2.2. (see [20]) As defined above (Ω,F,P,{θt}t∈R) is the metric dynamical system and the Ornstein-Uhlenbeck process {z(θtω)}t∈R. Then one has a θt-invariant set ˜Ω∈Ω of full P-measure such that the following statements are satisfied
(ⅰ) The Ornstein-Uhlenbeck process {z(θtω)}t∈R has the asymptotically sublinear growth property, i.e.,
limt→±∞|z(θtω)||t|=0, | (2.10) |
(ⅱ) z(θtω) is continuous with respect to t, and
limt→±∞1t−t0∫tt0z(θsω)ds=0,∀t0∈R. | (2.11) |
Noticing that we consider ω∈˜Ω only and will always write Ω for ˜Ω.
In order to study the random dynamics of the stochastic extended Brusselator system, we usually transform stochastic Eqs. (1.1)–(1.6) to a system of pathwise PDEs.
Let
U=e−ρz(θtω)u,V=e−ρz(θtω)v,Φ=e−ρz(θtω)φ,W=e−ρz(θtω)w,Z=e−ρz(θtω)z,Ψ=e−ρz(θtω)ψ, | (2.12) |
here z(θtω) is the Ornstein-Uhlenbeck process as above in (2.8). Therefore, we have
dU=−ρe−ρz(θtω)u∘dz+e−ρz(θtω)du,dV=−ρe−ρz(θtω)v∘dz+e−ρz(θtω)dv,dΦ=−ρe−ρz(θtω)φ∘dz+e−ρz(θtω)dφ,dW=−ρe−ρz(θtω)w∘dz+e−ρz(θtω)dw,dZ=−ρe−ρz(θtω)z∘dz+e−ρz(θtω)dz,dΨ=−ρe−ρz(θtω)ψ∘dz+e−ρz(θtω)dψ. |
By using (2.9), we convert Eqs. (1.1)–(1.6) to a random PDE system
{dUdt=d1ΔU+ae−ρz(θtω)−(b+k)U+e2ρz(θtω)U2V+D1(W−U)+NΦ+ρz(θtω)U,(2.13)dVdt=d2ΔV+bU−e2ρz(θtω)U2V+D2(Z−V)+ρz(θtω)V,(2.14)dΦdt=d3ΔΦ+kU−(λ+N)Φ+D3(Ψ−Φ)+ρz(θtω)Φ,(2.15)dWdt=d1ΔW+ae−ρz(θtω)−(b+k)W+e2ρz(θtω)W2Z+D1(U−W)+NΨ+ρz(θtω)W,(2.16)dZdt=d2ΔZ+bW−e2ρz(θtω)W2Z+D2(V−Z)+ρz(θtω)Z,(2.17)dΨdt=d3ΔΨ+kW−(λ+N)Ψ+D3(Φ−Ψ)+ρz(θtω)Ψ,(2.18) |
for all ω∈Ω, x∈O and t>t0, with boundary conditions
U(t,ω,x)=V(t,ω,x)=Φ(t,ω,x)=0,t>t0∈R,W(t,ω,x)=Z(t,ω,x)=Ψ(t,ω,x)=0,x∈∂O,ω∈Ω, |
and the initial conditions
U(t0,ω,x)=U0(ω,x)=e−ρz(θt0ω)u0(x),V(t0,ω,x)=V0(ω,x)=e−ρz(θt0ω)v0(x),Φ(t0,ω,x)=Φ0(ω,x)=e−ρz(θt0ω)φ0(x),W(t0,ω,x)=W0(ω,x)=e−ρz(θt0ω)w0(x),Z(t0,ω,x)=Z0(ω,x)=e−ρz(θt0ω)z0(x),Ψ(t0,ω,x)=Ψ0(ω,x)=e−ρz(θt0ω)ψ0(x). |
We can express the Eqs. (2.13)–(2.18) as
dgdt=Ag+F(g,θtω),∀ω∈Ω, | (2.19) |
g(t0,ω,x)=g0=(U0(ω,x),V0(ω,x),Φ0(ω,x),W0(ω,x),Z0(ω,x),Ψ0(ω,x))T, | (2.20) |
where
F(g,θtω)=(ae−ρz(θtω)−(b+k)U+e2ρz(θtω)U2V+D1(W−U)+NΦ+ρz(θtω)UbU−e2ρz(θtω)U2V+D2(Z−V)+ρz(θtω)VkU−(λ+N)Φ+D3(Ψ−Φ)+ρz(θtω)Φae−ρz(θtω)−(b+k)W+e2ρz(θtω)W2Z+D1(U−W)+NΨ+ρz(θtω)WbW−e2ρz(θtω)W2Z+D2(V−Z)+ρz(θtω)ZkW−(λ+N)Ψ+D3(Φ−Ψ)+ρz(θtω)Ψ). |
Similar to deterministic system, we prove the local existence and uniqueness of the weak solution g(t,ω;t0,g0),t∈[t0,T(ω,g0)] for some T(ω,g0)>t0 by the Galerkin approximations and compactness argument[27]. According to the parabolic regularity theory in [26], each weak solution will become a strong solution for t>t0 in the existence interval. Integrating with Lemma 2.2 in [15], the weak solution g(t,ω;t0,g0) of the random extended Brusselator evolutionary system (2.19)–(2.20) on the maximal existence time interval, which satisfies
g(t,ω;t0,g0)∈C([t0,Tmax);H)∩C1((t0,Tmax);H)∩L2((t0,Tmax);E). |
Therefore, we need to study that the global existence and uniqueness of the weak solutions for the extended Brusselator random dynamic system (2.19)–(2.20) in the next section.
Then, we find that the system (2.6)–(2.7) generate a continuous RDS f:R+×Ω×H→H over MDS, which satisfies
f(t−τ,θτω,f0)=S(t,τ,ω)f0=eρz(θtω)g(t,ω;τ,g0),∀t≥τ,ω∈Ω. | (2.21) |
Owing to (2.21), the following pullback relation is established
f(t,θ−tω,f0)=eρz(ω)g(0,ω;−t,eρz(θ−tω)f0),fort≥0, | (2.22) |
which be called the pullback quasi-trajectory from g0. We will study the pullback asymptotic behavior by establishing the pullback quasi-trajectory.
In this section, we firstly prove the existence and uniqueness of the weak solution by applying the scaling method and estimate groups. Then, we obtain the pullback absorbing property and some necessary estimates. For convenience, the U(t,ω;t0,U0), V(t,ω;t0,V0) and Φ(t,ω;t0,Φ0) et al are shorthand for U(t,ω), V(t,ω) and Φ(t,ω) or U, V and Φ.
Lemma 3.1. Let R(ω)>0 be a given tempered random variable and for every initial value f0=(u0,v0,φ0,w0,z0,ψ0)∈H, where ||f0||≤R(ω), then one has a time variable T(R,ω)≤−1 such that for all initial time t0≤T(R,ω), the weak solution g(t,ω)=(U(t,ω),V(t,ω),Φ(t,ω),W(t,ω),Z(t,ω),Ψ(t,ω)) of the random extended Brusselator equations (2.13)–(2.18) exists on [t0,0].
Furthermore, suppose t0≤min{T(R,ω),−4}, then one has a random variable M(t,ω) for terminal time t∈[−4,0] such that the weak solution g satisfies the following inequality
||g(t,ω;t0,e−ρz(θt0ω)f0)||2≤M(t,ω),∀t≥t0,ω∈Ω. | (3.1) |
Proof. We take the scalar products (2.14) with V(t) and (2.17) with Z(t), and then add them up, which follows that
12ddt(||V||2+||Z||2)+d2(||∇V||2+||∇Z||2)=−e2ρz(θtω)∫O[(UV−12be−2ρz(θtω))2+(WZ−12be−2ρz(θtω))2]dx−∫OD2(V−Z)2dx+12b2|O|e−2ρz(θtω)+ρz(θtω)(||V||2+||Z||2)≤12b2|O|e−2ρz(θtω)+ρz(θtω)(||V||2+||Z||2). | (3.2) |
Applying Poincˊare inequality, we get
ddt(||V||2+||Z||2)+2γd2(||V||2+||Z||2)≤ddt(||V||2+||Z||2)+2d2(||∇V||2+||∇Z||2)≤b2|O|e−2ρz(θtω)+2ρz(θtω)(||V||2+||Z||2), | (3.3) |
multiplying both sides of (3.3) by e∫tt0(2ρz(θsω)−2γd2)ds and integrating over the time interval [t0,t], where t0<−4<t<0, which yields
||V(t,ω;t0,g0)||2+||Z(t,ω;t0,g0)||2≤(||V0||2+||Z0||2)e∫tt02ρz(θsω)ds−2γd2(t−t0)+b2|O|∫tt0e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ. | (3.4) |
Then, we take the pullback estimate by asymptotic decay of the Ornstein-Uhlenbeck process, which get rid of the dependence on the initial value and time. This play an important role in this paper.
According to (2.10) and (2.11), for any random variable R(ω)>0, one has a time variable T1(R,ω)<−4 such that for each t0≤T1(R,ω) and t∈[−4,0], which follows that
{1t−t0∫tt06ρz(θsω)ds−γd′≤−12γd′,(3.5)e−12γd′(t−t0)e−ρz(θt0ω)R2(ω)≤1,(3.6) |
where d′=min{d1,d2,d3}. Applying (3.5) and (3.6), one has T2(R,ω)<−4 such that for all τ≤T2(R,ω), we infer that
e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)=e(t−τ)(∫tτ2ρz(θsω)dst−τ−2γd2−2ρz(θτω)t−τ)≤e−12γd′(t−τ), | (3.7) |
and then
∫τ−∞e−12γd′(t−σ)dσ≤∫T2−∞e−12γd′(t−σ)dσ=2γd′e12γd′(T2−t). | (3.8) |
Therefore, we obtain that
∫t−∞e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ | (3.9) |
is convergent. In this way, we get
||V(t,ω;t0,g0)||2+||Z(t,ω;t0,g0)||2≤(||V0||2+||Z0||2)e∫tt02ρz(θsω)ds−2γd2(t−t0)+b2|O|∫tt0e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ≤1+b2|O|∫t−∞e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ. | (3.10) |
Let y(t,x,ω)=U(t,x,ω)+V(t,x,ω)+W(t,x,ω)+Z(t,x,ω) and ξ(t,x,ω)=Φ(t,x,ω)+Ψ(t,x,ω), then by applying (2.13)–(2.18), we deduce that
dydt=d1Δy−ky+[(d2−d1)Δ(V+Z)+k(V+Z)+2ae−ρz(θtω)]+Nξ+ρz(θtω)y, | (3.11) |
dξdt=d3Δξ+ky−k(V+Z)−(λ+N)ξ+ρz(θtω)ξ. | (3.12) |
Rescaling ξ=μη with μ=k/N, we have
dydt=d1Δy−ky+[(d2−d1)Δ(V+Z)+k(V+Z)+2ae−ρz(θtω)]+kη+ρz(θtω)y, | (3.13) |
μdηdt=μd3Δη+ky−k(V+Z)−(μλ+k)η+μρz(θtω)η. | (3.14) |
Then, we take the inner product (3.13) with y(t) and apply Poincˊare inequality, we get
12ddt||y||2+d1||∇y||2≤∫O[(d2−d1)Δ(V+Z)+k(V+Z)+2ae−ρz(θtω)]ydx−k||y||2+k||η||||y||+ρz(θtω)||y||2≤|d1−d2|||∇(V+Z)||||∇y||+k||V+Z||||y||+2ae−ρz(θtω)|O|12||y||−k||y||2+k||η||||y||+ρz(θtω)||y||2≤d14||∇y||2+|d1−d2|2d1||∇(V+Z)||2+2k2d1γ||V+Z||2+d1γ4||y||2+8d1γa2|O|e−2ρz(θtω)−k||y||2+k||η||||y||+ρz(θtω)||y||2≤d12||∇y||2+|d1−d2|2d1||∇(V+Z)||2+2k2d1γ||V+Z||2+8d1γa2|O|e−2ρz(θtω)−k||y||2+k||η||||y||+ρz(θtω)||y||2, | (3.15) |
so, we obtain that
ddt||y||2+d1||∇y||2≤2|d1−d2|2d1||∇(V+Z)||2+4k2d1γ||V+Z||2+16d1γa2|O|e−2ρz(θtω)−2k||y||2+2k||η||||y||+2ρz(θtω)||y||2. | (3.16) |
Taking the inner product (3.14) with η(t), we have
12μddt||η||2+μd3||∇η||2≤k||y||||η||+k||V+Z||||η||−(μλ+k)||η||2+μρz(θtω)||η||2≤k||y||||η||+μγd32||η||2+k22μγd3||V+Z||2−(μλ+k)||η||2+μρz(θtω)||η||2, | (3.17) |
we deduce that
μddt||η||2+μd3||∇η||2≤2k||y||||η||+k2μγd3||V+Z||2−2(μλ+k)||η||2+2μρz(θtω)||η||2. | (3.18) |
Adding (3.16) and (3.18) up, and noticing that
−2k||y||2+4k||η||||y||−2(μλ+k)||η||2≤0, |
so that
ddt(||y||2+||μ−12ξ||2)+min{d1,d3}(||∇y||2+||μ−12∇ξ||2)≤2|d1−d2|2d1||∇(V+Z)||2+(4k2d1γ+k2μγd3)||V+Z||2+2ρz(θtω)(||y||2+||μ−12ξ||2)+16d1γa2|O|e−2ρz(θtω). | (3.19) |
Let d=min{d1,d3}, α=max{1,μ−1}min{1,μ−1}. Then, multiplying both sides of (3.19) by e∫tt0(2ρz(θsω)−γd)ds and integrating over [t0,t], where t0<−4<t<0. Therefore, one has a time variable T3(R,ω)<−4 such that for each t0≤T3(R,ω) and t∈[−4,0], we obtain that
||y(t,ω;t0,g0)||2+||ξ(t,ω;t0,g0)||2≤α(||y0||2+||ξ0||2)e∫tt02ρz(θsω)ds−γd(t−t0)+α∫tt0e∫tτ(2ρz(θsω)−γd)ds[2|d1−d2|2d1||∇V(τ)+∇Z(τ)||2]dτ+α∫tt0e∫tτ(2ρz(θsω)−γd)ds[(4k2d1γ+k2μγd3)||V(τ)+Z(τ)||2]dτ+α∫tt0e∫tτ(2ρz(θsω)−γd)ds(16a2d1γ|O|e−2ρz(θτω))dτ. | (3.20) |
By applying Poincˊare inequality and above estimates (3.10), it follows that
||y(t,ω;t0,g0)||2+||ξ(t,ω;t0,g0)||2≤α+α∫tt0e∫tτ(2ρz(θsω)−γd)ds[β(||∇V(τ)||2+||∇Z(τ)||2)+16a2d1γ|O|e−2ρz(θτω)]dτ, | (3.21) |
where β=4|d1−d2|2d1+8k2d1γ2+2k2μd3γ2.
Then, we multiply both sides of (3.3) by e∫tt0(2ρz(θsω)−γd)ds and integrate over [t0,t], where t0<−4<t<0, one has a time variable T4(R,ω)<−4 such that
2d2∫tt0e∫tτ(2ρz(θsω)−γd)ds(||∇V(τ,ω;t0,g0)||2+||∇Z(τ,ω;t0,g0)||2)dτ≤e∫tτ(2ρz(θsω)−γd)ds(||V(t0)||2+||Z(t0)||2)+∫tt0(||V(τ)||2+||Z(τ)||2)(−2ρz(θτω)+γd)e∫tτ(2ρz(θsω)−γd)dsdτ+∫tt0e∫tτ(2ρz(θsω)−γd)ds[2ρz(θτω)(||V(τ)||2+||Z(τ)||2)+b2|O|e−2ρz(θτω)]dτ≤1+b2|O|∫tt0e∫tτ(2ρz(θsω)−γd)dse−2ρz(θτω)dτ+γd∫tt0(||V(τ)||2+||Z(τ)||2)e∫tτ(2ρz(θsω)−γd)dsdτ. | (3.22) |
Now, we deal with the last integral term in (3.22). Multiplying both sides of (3.4) by e∫tt0(2ρz(θsω)−γd)ds, and integrating over [t0,t], where t0<−4<t<0. Then, one has a time variable T5(R,ω)<T1(R,ω) such that for every t0≤T5(ω), it follows that
∫tt0e∫tτ(2ρz(θsω)−γd)ds(||V(τ,ω;t0,g0)||2+||Z(τ,ω;t0,g0)||2)dτ≤∫tt0e∫tτ(2ρz(θsω)−γd)ds[(||V0||2+||Z0||2)e∫τt02ρz(θsω)ds−2γd2(τ−t0)]dτ+b2|O|∫tt0e∫tτ(2ρz(θsω)−γd)ds∫τt0e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξdτ≤∫tt0e∫tt0(2ρz(θsω)−γd′)ds(||V0||2+||Z0||2)dτ+b2|O|∫tt0∫tξe∫tτ(2ρz(θsω)−γd)dse∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dτdξ≤(t−t0)(||V0||2+||Z0||2)e∫tt0(2ρz(θsω)−γd′)ds+b2|O|∫tt0∫tξe∫tξ2ρz(θsω)ds−∫tξγd′ds−2ρz(θξω)dτdξ=(t−t0)(||V0||2+||Z0||2)e∫tt0(2ρz(θsω)−γd′)ds+b2|O|∫tt0(t−ξ)e∫tξ2ρz(θsω)ds−∫tξγd′ds−2ρz(θξω)dξ≤1+b2|O|∫t−∞(t−ξ)e∫tξ2ρz(θsω)ds−∫tξγd′ds−2ρz(θξω)dξ, | (3.23) |
where d′=min{d1,d2,d3}. Substituting (3.23) into (3.22), we deduce
C1(t,ω)=∫tt0e∫tτ(2ρz(θsω)−γd)ds(||∇V(τ,ω;t0,g0)||2+||∇Z(τ,ω;t0,g0)||2)dτ=12d2+b2|O|2d2∫tt0e∫tτ(2ρz(θsω)−γd)dse−2ρz(θτω)dτ+γd2d2[1+b2|O|∫t−∞(t−ξ)e∫tξ2ρz(θsω)ds−∫tξγd′ds−2ρz(θξω)dξ] | (3.24) |
is tempered by (2.10) and (2.11). Then, substituting (3.24) into (3.21), we obtain that for t0≤min{T3(R,ω),T4(R,ω),T5(R,ω)}
||y(t,ω;t0,g0)||2+||ξ(t,ω;t0,g0)||2≤α+αβC1(t,ω)+16a2|O|αd1γ∫t−∞e∫tτ(2ρz(θsω)−γd)ds−2ρz(θτω)dτ=C2(t,ω). | (3.25) |
Owing to (3.4) and (3.25), we get
||U(t,ω;t0,g0)+W(t,ω;t0,g0)||2+||Φ(t,ω;t0,g0)+Ψ(t,ω;t0,g0)||2=||y(t,ω;t0,g0)−(V(t,ω;t0,g0)+Z(t,ω;t0,g0))||2+||Φ(t,ω;t0,g0)+Ψ(t,ω;t0,g0)||2≤2||y(t,ω;t0,g0)||2+4(||V(t,ω;t0,g0)||2+||Z(t,ω;t0,g0)||2)+||ξ(t,ω;t0,g0)||2≤2C2(t,ω)+4(1+b2|O|∫tt0e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ). | (3.26) |
Next, we deal with the other components. Let p(t,x,ω)=U(t,x,ω)+V(t,x,ω)−W(t,x,ω)−Z(t,x,ω) and q(t,x,ω)=Φ(t,x,ω)−Ψ(t,x,ω), we have
dpdt=d1Δp−(k+2D1)p+(d2−d1)Δ(V−Z)+[k+2(D1−D2)](V−Z)+Nq+ρz(θtω)p, | (3.27) |
dqdt=d3Δq+kp−k(V−Z)−(λ+N)q−2D3q+ρz(θtω)q. | (3.28) |
Rescaling q=μϖ with μ=k/N, we have
dpdt=d1Δp−(k+2D1)p+(d2−d1)Δ(V−Z)+[k+2(D1−D2)](V−Z)+kϖ+ρz(θtω)p, | (3.29) |
μdϖdt=μd3Δϖ+kp−k(V−Z)−(μλ+k)ϖ−2μD3ϖ+μρz(θtω)ϖ. | (3.30) |
Taking the inner product (3.29) with p(t), then using Hölder inequality and Young inequality, we obtain
12ddt||p||2+d1||∇p||2+(k+2D1)||p||2≤|d1−d2|||∇(V−Z)||||∇p||+[k+2(D1−D2)]||V−Z||||p||+k||ϖ||||p||+ρz(θtω)||p||2≤d12||∇p||2+|d1−d2|22d1||∇(V−Z)||2+|k+2(D1−D2)|28D1||V−Z||2+2D1||p||2+k||ϖ||||p||+ρz(θtω)||p||2, | (3.31) |
thus, we get
ddt||p||2+d1||∇p||2≤|d1−d2|2d1||∇(V−Z)||2+|k+2(D1−D2)|24D1||V−Z||2−2k||p||2+2k||ϖ||||p||+2ρz(θtω)||p||2. | (3.32) |
Next, taking the inner products of (3.30) with ϖ(t), we have
12μddt||ϖ||2+μd3||∇ϖ||2≤k||p||||ϖ||+k||V−Z||||ϖ||−(μλ+k+2μD3)||ϖ||2+μρz(θtω)||ϖ||2≤2μD3||ϖ||2+k28μD3||V−Z||2+k||p||||ϖ||−(μλ+k+2μD3)||ϖ||2+μρz(θtω)||ϖ||2≤k24μD3(||V||2+||Z||2)+k||p||||ϖ||−(μλ+k)||ϖ||2+μρz(θtω)||ϖ||2, | (3.33) |
thus, we obtain
μddt||ϖ||2+μd3||∇ϖ||2≤k22μD3(||V||2+||Z||2)+2k||p||||ϖ||−2(μλ+k)||ϖ||2+2μρz(θtω)||ϖ||2. | (3.34) |
Adding (3.32) and (3.34) up, and noticing that
−2k||p||2+4k||ϖ||||p||−2(μλ+k)||ϖ||2≤0. | (3.35) |
Therefore, applying the Poincˊare inequality, we have
ddt(||p||2+||μ−12q||2)+min{d1,d3}(||∇p||2+||μ−12∇q||2)≤2|d1−d2|2d1(||∇V||2+||∇Z||2)+(|k+2(D1−D2)|22D1+k22μD3)(||V||2+||Z||2)+2ρz(θtω)(||p||2+||μ−12q||2)≤κ(||∇V||2+||∇Z||2)+2ρz(θtω)(||p||2+||μ−12q||2), | (3.36) |
where κ=2|d1−d2|2d1+|k+2(D1−D2)|22γD1+k22γμD3.
Multiplying both sides of (3.36) by e∫tt0(2ρz(θsω)−γd)ds and integrating over [t0,t], where t0<−4<t<0. Therefore, one has a time variable T6(R,ω)<−4 such that for each t0≤T6(R,ω) and t∈[−4,0], by using (3.24), we obtain that
||p(t,ω;t0,g0)||2+||q(t,ω;t0,g0)||2≤α(||p0||2+||q0||2)e∫tt0(2ρz(θsω)−γd)ds+ακ∫tt0e∫tτ(2ρz(θsω)−γd)ds(||∇V||2+||∇Z||2)dτ=C3(t,ω), | (3.37) |
where C3(t,ω)=α+ακC1(t,ω). From (3.4), (3.24) and (3.37), we infer that
||U(t,ω;t0,g0)−W(t,ω;t0,g0)||2+||Φ(t,ω;t0,g0)−Ψ(t,ω;t0,g0)||2=||p(t,ω)−(V(t,ω)−Z(t,ω))||2+||Φ(t,ω)−Ψ(t,ω)||2≤2||p(t,ω)||2+4(||V(t,ω)||2+||Z(t,ω)||2)+||q(t,ω)||2≤2C3(t,ω)+4(1+b2|O|∫tt0e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ). | (3.38) |
By using (3.10), (3.26) and (3.38), we have
U(t,ω)=12[(U(t,ω)+W(t,ω))+(U(t,ω)−W(t,ω))], | (3.39) |
W(t,ω)=12[(U(t,ω)+W(t,ω))−(U(t,ω)−W(t,ω))], | (3.40) |
Φ(t,ω)=12[(Φ(t,ω)+Ψ(t,ω))+(Φ(t,ω)−Ψ(t,ω))], | (3.41) |
Ψ(t,ω)=12[(Φ(t,ω)+Ψ(t,ω))−(Φ(t,ω)−Ψ(t,ω))] | (3.42) |
is uniformly bounded.
Let T(R,ω)=min{T1(R,ω),T2(R,ω),T3(R,ω),T4(R,ω),T5(R,ω),T6(R,ω)}. Then, for t0≤T(R,ω) and t∈[−4,0], we obtain that
||g(t,ω;t0,g0)||2=||U(t,ω)||2+||V(t,ω)||2+||Φ(t,ω)||2+||W(t,ω)||2+||Z(t,ω)||2+||Ψ(t,ω)||2≤14||(U+W)+(U−W)||2+14||(U+W)−(U−W)||2+14||(Φ+Ψ)+(Φ−Ψ)||2+14||(Φ+Ψ)−(Φ−Ψ)||2+||V||2+||Z||2≤||U+W||2+||U−W||2+||Φ+Ψ||2+||Φ−Ψ||2+||V||2+||Z||2≤2C2(t,ω)+2C3(t,ω)+9(1+b2|O|∫tt0e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ)=M(t,ω). | (3.43) |
In this way, we have proved the global existence and uniqueness of the weak solution.
Since g(t,ω;τ,g0) is the weak solution to Eqs. (2.13)–(2.18), then
f(t,ω;τ,f0)=S(t,τ,ω)f0=eρz(θtω)g(t,ω;τ,g0),t≥τ, | (3.44) |
is the solution to Eqs. (1.1)–(1.9), where
f0=(u0,v0,φ0,w0,z0,ψ0),g0=e−ρz(θτω)f0. | (3.45) |
Lemma 3.2. For the extended Brusselator random dynamical system f on H over the MDS, one has a D-pullback absorbing set B0(ω), where B0(ω) is the random ball centered at the origin with the radius M0(ω) given by
M0(ω)=eρz(θtω)[2C2(0,ω)+2C3(0,ω)]+9eρz(θtω)(1+b2|O|∫0−∞e∫0τ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ). | (3.46) |
Proof. According to the Lemma 3.1, we can obtain the consequence of Lemma 3.2. The more details of the proof, see, e.g., [16].
If we want to prove the pullback asymptotic compactness in the next section, we have to establish some necessary estimates for the V-component and Z-component in L6(O). Based on this, the following Lemma is given.
Lemma 3.3. For every given initial value f0∈E, for the terminal time t∈[−4,0], one has a random variable P(t,ω)>0 such that for any initial time t0≤T7(||g0||L6,ω)≤−4, we obtain that
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤P(t,ω),−4≤t≤0. | (3.47) |
Proof. We take the scalar product (2.14) with V3 and (2.17) with Z3, and add them up, then by using Young inequality, it follows that
14ddt(||V||4+||Z||4)+3d2(||V(t)∇V(t)||2+||Z(t)∇Z(t)||2)=∫O[bUV3+bWZ3−e2ρz(θtω)(U2V4+W2Z4)]dx+D2∫O[(Z−V)V3+(V−Z)Z3]dx+∫Oρz(θtω)(V4+Z4)dx≤∫O[12b2e−2ρz(θtω)(V2+Z2)+12e2ρz(θtω)(U2V4+W2Z4)]dx−∫Oe2ρz(θtω)(U2V4+W2Z4)dx+∫Oρz(θtω)(V4+Z4)dx, | (3.48) |
where
∫O[(Z−V)V3+(V−Z)Z3]dx≤∫O[−V4+(14Z4+34V4)+(14V4+34Z4)−Z4]dx=0. |
Therefore, we get
ddt(||V(t)||4L4+||Z(t)||4L4)+3γd2(||V(t)||4L4+||Z(t)||4L4)≤ddt(||V(t)||4L4+||Z(t)||4L4)+3d2(||∇V2(t)||2+||∇Z2(t)||2)≤2b2e−2ρz(θtω)(||V(t)||2+||Z(t)||2)+4ρz(θtω)(||V(t)||4L4+||Z(t)||4L4). | (3.49) |
We multiply both sides of (3.49) by e∫tt0(4ρz(θsω)−3γd2)ds, and integrate over the interval [t0,t], here t0<t. Then by applying (3.4), we obtain that
||V(t,ω;t0,g0)||4L4+||Z(t,ω;t0,g0)||4L4≤(||V0||4L4+||Z0||4L4)e∫tt0(4ρz(θsω)−3γd2)ds+2b2∫tt0e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)(||V(τ,ω)||2+||Z(τ,ω)||2)dτ≤(||V0||4L4+||Z0||4L4)e∫tt0(4ρz(θsω)−3γd2)ds+2b2∫tt0e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)(||V0||2+||Z0||2)e∫τt0(2ρz(θsω)−2γd2)dsdτ+2b4|O|∫tt0e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)∫τt0e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξdτ=(||V0||4L4+||Z0||4L4)e∫tt0(4ρz(θsω)−3γd2)ds+2b2(||V0||2+||Z0||2)∫tt0e∫tt0(4ρz(θsω)−2γd2)ds−2ρz(θτω)dτ+2b4|O|∫tt0∫tξe−2ρz(θτω)e∫tτ(4ρz(θsω)−3γd2)dse∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dτdξ≤(||V0||4L4+||Z0||4L4)e∫tt0(4ρz(θsω)−3γd2)ds+2b2(||V0||2+||Z0||2)∫tt0e∫tt0(4ρz(θsω)−2γd2)ds−2ρz(θτω)dτ+2b4|O|∫tt0∫tξe−2ρz(θτω)e∫tξ(4ρz(θsω)−2γd2)ds−2ρz(θξω)dτdξ≤(||V0||4L4+||Z0||4L4)e∫tt0(4ρz(θsω)−3γd2)ds+2b2(||V0||2+||Z0||2)e∫tt0(4ρz(θsω)−2γd2)ds∫tt0e−2ρz(θτω)dτ+2b4|O|∫tt0∫tt0e−2ρz(θτω)e∫tξ(4ρz(θsω)−2γd2)ds−2ρz(θξω)dτdξ. | (3.50) |
Then, taking the inner product of (2.14) with V5 and (2.17) with Z5, we get
16ddt(||V||6+||Z||6)+5d2(||V2(t)∇V(t)||2+||Z2(t)∇Z(t)||2)=∫O[bUV5−e2ρz(θtω)U2V6+bWZ5−e2ρz(θtω)W2Z6]dx+∫O[D2(Z−V)V5+D2(V−Z)Z5]dx+∫Oρz(θtω)(V6+Z6)dx, | (3.51) |
by using Young inequality and H¨older inequality, we have
∫O[bUV5−e2ρz(θtω)U2V6+bWZ5−e2ρz(θtω)W2Z6]dx≤∫O[12b2e−2ρz(θtω)(V4+Z4)−12e2ρz(θtω)(U2V6+W2Z6)]dx≤∫O12b2e−2ρz(θtω)(V4+Z4)dx, | (3.52) |
and
∫O[(Z−V)V5+(V−Z)Z5]dx≤∫O[−V6+(16Z6+56V6)+(16V6+56Z6)−Z6]dx=0. | (3.53) |
Substituting (3.52) and (3.53) into (3.51), and then applying Poincˊare inequality, we obtain that
ddt(||V(t)||6+||Z(t)||6)+3γd2(||V(t)||6+||Z(t)||6)≤ddt(||V(t)||6+||Z(t)||6)+103d2(||∇V3(t)||2+||∇Z3(t)||2)≤3b2e−2ρz(θtω)(||V(t)||4+||Z(t)||4)+6ρz(θtω)(||V(t)||6+||Z(t)||6). | (3.54) |
We multiply both sides of (3.54) by e∫tt0(6ρz(θsω)−3γd2)ds, and integrate over [t0,t], here t0<−4<t<0. Then one has a random variable T7(||g0||L6,ω)≤−4 such that for any ω∈Ω, t0≤T7(||g0||L6,ω) and t∈[−4,0], by applying (3.50), we obtain that
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤(||V0||6L6+||Z0||6L6)e∫tt0(6ρz(θsω)−3γd2)ds+3b2∫tt0e∫tη(6ρz(θsω)−3γd2)ds−2ρz(θηω)(||V(η,ω)||4L4+||Z(η,ω)||4L4)dη≤(||V0||6L6+||Z0||6L6)e∫tt0(6ρz(θsω)−3γd2)ds+3b2∫tt0e∫tη(6ρz(θsω)−3γd2)ds−2ρz(θηω)(||V0||4L4+||Z0||4L4)e∫ηt0(4ρz(θsω)−3γd2)dsdη+6b4(||V0||2+||Z0||2)∫tt0e∫tη(6ρz(θsω)−3γd2)ds−2ρz(θηω)e∫ηt0(4ρz(θsω)−2γd2)ds⋅∫ηt0e−2ρz(θτω)dτdη+6b6|O|∫tt0e∫tη(6ρz(θsω)−3γd2)ds−2ρz(θηω)∫ηt0e−2ρz(θτω)dτ⋅∫ηt0e∫ηξ(4ρz(θsω)−2γd2)ds−2ρz(θξω)dξdη≤(||V0||6L6+||Z0||6L6)e∫tt0(6ρz(θsω)−3γd2)ds+3b2(||V0||4L4+||Z0||4L4)∫tt0e−2ρz(θηω)e∫tt0(6ρz(θsω)−3γd2)dsdη+6b4(||V0||2+||Z0||2)∫tt0e−2ρz(θηω)e∫tt0(6ρz(θsω)−2γd2)ds∫ηt0e−2ρz(θτω)dτdη+6b6|O|∫tt0e−2ρz(θτω)dτ∫tt0∫ηt0e∫tη(6ρz(θsω)−3γd2)ds−2ρz(θηω)⋅e∫ηξ(4ρz(θsω)−2γd2)ds−2ρz(θξω)dξdη≤(||V0||6L6+||Z0||6L6)e∫tt06ρz(θsω)ds−3γd2(t−t0)+3b2(||V0||4L4+||Z0||4L4)e∫tt0(6ρz(θsω)−3γd2)ds∫tt0e−2ρz(θηω)dη+6b4(||V0||2+||Z0||2)e∫tt0(6ρz(θsω)−2γd2)ds∫tt0e−2ρz(θηω)∫ηt0e−2ρz(θτω)dτdη+6b6|O|∫tt0e−2ρz(θτω)dτ∫tt0∫tξe−2ρz(θηω)e∫tξ(6ρz(θsω)−2γd2)ds−2ρz(θξω)dηdξ≤(||V0||6L6+||Z0||6L6)e∫tt06ρz(θsω)ds−3γd2(t−t0)+3b2(||V0||4L4+||Z0||4L4)e∫tt0(6ρz(θsω)−3γd2)ds∫tt0e−2ρz(θηω)dη+6b4(||V0||2+||Z0||2)e∫tt0(6ρz(θsω)−2γd2)ds∫tt0e−2ρz(θηω)dη∫tt0e−2ρz(θτω)dτ+6b6|O|∫tt0e−2ρz(θτω)dτ∫tt0e−2ρz(θηω)dη∫tt0e∫tξ(6ρz(θsω)−2γd2)ds−2ρz(θξω)dξ. | (3.55) |
Therefore, we get
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤P(t,ω), | (3.56) |
where
P(t,ω)=3+6b6|O|∫tt0e−2ρz(θτω)dτ∫tt0e−2ρz(θηω)dη∫tt0e∫tξ(6ρz(θsω)−2γd2)ds−2ρz(θξω)dξ. |
Owing to (3.5)–(3.8), we deduce that P(t,ω) is convergent. Therefore, we have completed the proof.
Lemma 3.4. Let R(ω)>0 be a given tempered random variable and for all t0<−4<t1<0 and every initial value f0∈H, where ||f0||≤R(ω). Suppose that g(t,ω;t0,g0) satisfies ||g(t1,ω;t0,g0)||∈E with
||g(t1,ω;t0,g0)||E≤G(ω), |
here G(ω)>0 is any given random variable. For all t∈[t1,0], then one has a random variable D(t,G,ω)>0 such that
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤D(t,G,ω),∀t0≤min{T(R,ω),−4}. | (3.57) |
Proof. For any initial time t0≤min{T(R,ω),−4}. We integrate (3.49) over [t1,t] and apply (3.10), which follows that
||V(t,ω;t0,g0)||4L4+||Z(t,ω;t0,g0)||4L4≤(||V(t1,ω;t0,g0)||4L4+||Z(t1,ω;t0,g0)||4L4)e∫tt1(4ρz(θsω)−3γd2)ds+2b2∫tt1e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)(||V(τ,ω)||2+||Z(τ,ω)||)2dτ≤δ4G4(ω)e∫tt1(4ρz(θsω)−3γd2)ds+2b2∫tt1e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)dτ+2b4|O|∫tt1e∫tτ(4ρz(θsω)−3γd2)ds−2ρz(θτω)∫τ−∞e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξdτ≤Π(t,ω), | (3.58) |
where δ is the constant of the Sobolev embedding H10(O)↪L4(O) satisfies
||ϕ||L4(O)≤δ||ϕ||E,∀ϕ∈E. |
For each initial time t0≤min{T(R,ω),−4}, we integrate (3.54) over [t1,t], then by using (2.4) and (3.58), we obtain that
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤(||V(t1,ω;t0,g0)||6L6+||Z(t1,ω;t0,g0)||6L6)e∫tt1(6ρz(θsω)−3γd2)ds+3b2∫tt1e∫tτ(6ρz(θsω)−3γd2)ds−2ρz(θτω)(||V(τ,ω)||4L4+||Z(τ,ω)||4L4)dτ≤ζ6G6(ω)e∫tt1(6ρz(θsω)−3γd2)ds+3b2∫tt1e∫tτ(6ρz(θsω)−3γd2)ds−2ρz(θτω)Π(t,ω)dτ. | (3.59) |
Therefore,
D(t,G,ω)=ζ6G6(ω)e∫tt1(6ρz(θsω)−3γd2)ds+3b2∫tt1e∫tτ(6ρz(θsω)−3γd2)ds−2ρz(θτω)Π(t,ω)dτ. |
Then, we have completed the proof of Lemma 3.4.
In this section, we will apply the following uniform Gronwall inequality to study the pullback asymptotically compact of the extended Brusselator random dynamical system f in H, the reader can refer to reference [26] for more details.
Proposition 4.1. Assume n>1 is a given natural number. Let σ,π and χ be nonnegative functions in L1([−n,0];R+). Suppose that σ is absolutely continuous over [−n,0] and it satisfies the following inequality
dσdt≤πσ+χ,fort∈[−n,0]. |
If
∫t+1tπ(τ)dτ≤A,∫t+1tσ(τ)dτ≤B,∫t+1tχ(τ)dτ≤C,∀t∈[−n,−1], |
where A,B and C are some positive constants, then
σ(t)≤(B+C)eA,fort∈[−n+1,0]. |
Lemma 4.2. Let R(ω)>0 be a given random variable and for every initial value f0∈H, where ||f0||≤R(ω), one has a tempered random variable K(ω)>0 and a finite time variable T(R,ω)>0 such that for t0≤T(R,ω), the weak solution g(t,ω;τ,g0) of the random extended Brusselator Eqs. (2.13)–(2.18) satisfies g(0,ω;t0,g0)∈E and satisfies the following estimate
||g(0,ω;t0,g0)||2E≤K(ω),t0≤T(R,ω). | (4.1) |
Proof. We can divide into four steps to get the proof of Lemma 4.2.
Step 1. we establish the estimates of the H10(O)-norm for the U-component, V-component, Φ-component, W-component, Z-component and Ψ-component of the solution in [−4,−1].
Step 2. we study the estimates of the U-component and W-component in [−2,0] by applying the uniform Gronwall inequality.
Step 3. we obtain the estimates of the V-component and Z-component in [−1,0] by applying the results of the first and the second steps.
Step 4. we conduct the estimates of the Φ-component and Ψ-component in [−1,0].
Step 1. We study the time-average estimates of the E-norm for the weak solution g(t,ω)=(U,V,Φ,W,Z,Ψ). In Lemma 3.4, we have established the estimates of L6(O)-norm of the V-component and Z-component. Noticing that z(θtω) is continuous in t, then we obtain that for any given ω∈Ω, Z(ω)=max−4≤τ≤−1|z(θτω)| is a positive constant. We integrate (3.2) over [t,t+1], here −4≤t≤−1, then by applying (3.10), which follows that
∫t+1t2d2(||∇V(τ,ω;t0,g0)||2+||∇Z(τ,ω;t0,g0)||2)dτ≤||V(t)||2+||Z(t)||2+∫t+1tb2|O|e−2ρz(θτω)dτ+∫t+1t2ρz(θτω)(1+b2|O|∫τ−∞e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξ)dτ≤1+b2|O|max−4≤t≤−1∫t−∞e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ+b2|O|∫0−4e−2ρz(θτω)dτ+∫0−42c|z(θτω)|(1+b2|O|∫τ−∞e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξ)dτ. | (4.2) |
Then, for t0≤min{T(R,ω),−4} and −4≤t≤−1, we obtain that
∫t+1t(||∇V(τ,ω;t0,g0)||2+||∇Z(τ,ω;t0,g0)||2)dτ≤K1(ω)2d2, | (4.3) |
where
K1(ω)=1+b2|O|max−4≤t≤−1∫t−∞e∫tτ(2ρz(θsω)−2γd2)ds−2ρz(θτω)dτ+b2|O|∫0−4e−2ρz(θτω)dτ+∫0−42c|z(θτω)|(1+b2|O|∫τ−∞e∫τξ(2ρz(θsω)−2γd2)ds−2ρz(θξω)dξ)dτ. |
Let t=−4, which follows that
∫−3−4(||∇V(τ,ω;t0,g0)||2+||∇Z(τ,ω;t0,g0)||2)dτ≤K1(ω)2d2. | (4.4) |
According to the Mean Value Theorem, one has a time t1∈[−4,−3] such that
||V(t1,ω;t0,g0)||2E+||Z(t1,ω;t0,g0)||2E≤K1(ω)2d2. | (4.5) |
Therefore, by using Lemma 3.4, one has a random variable D(t,K12d1,ω)>0 such that
||V(t,ω;t0,g0)||6L6+||Z(t,ω;t0,g0)||6L6≤D(t,K12d1,ω),∀t∈[t1,0]. | (4.6) |
Integrating (3.19) over the interval [t,t+1], where −4≤t≤−1. For all t0≤min{T(R,ω),−4}, by using (3.25) and (4.5), we obtain that
d∫t+1t(||∇y(τ,ω;t0,g0)||2+||∇ξ(τ,ω;t0,g0)||2)dτ≤α(||y(t)||2+||ξ(t)||2)+αβ∫t+1t(||∇V(τ)||2+||∇Z(τ)||2)dτ+α∫t+1t2ρz(θτω)(||y(τ)||2+||ξ(τ)||2)dτ+16a2α|O|d1γ∫t+1te−2ρz(θτω)dτ≤αmax−4≤t≤−1C2(t,ω)+αβK1(ω)2d2+2αmax−4≤t≤−1C2(t,ω)∫0−4c|z(θτω)|dτ+16a2α|O|d1γ∫0−4e−2ρz(θτω)dτ. | (4.7) |
Therefore, we have
∫t+1t(||∇y(τ,ω;t0,g0)||2+||∇ξ(τ,ω;t0,g0)||2)dτ≤K2(ω)d, | (4.8) |
where
K2(ω)=αmax−4≤t≤−1C2(t,ω)+αβK1(ω)2d2+2αmax−4≤t≤−1C2(t,ω)∫0−4c|z(θτω)|dτ+16a2α|O|d1γ∫0−4e−2ρz(θτω)dτ. |
In this way, for any t0≤min{T(R,ω),−4} and −4≤t≤−1, we deduce that
∫t+1t(||∇U(τ,ω)+∇W(τ,ω)||2+||∇Φ(τ,ω)+∇Ψ(τ,ω)||2)dτ=∫t+1t(||∇y(τ,ω)−∇V(τ,ω)−∇Z(τ,ω)||2+||∇ξ(τ,ω)||2)dτ≤∫t+1t[2||∇y(τ,ω)||2+||∇ξ(τ,ω)||2+4(||∇V(τ,ω)||2+||∇Z(τ,ω)||2)]dτ≤2K2(ω)d+2K1(ω)d2=K3(ω). | (4.9) |
We integrate (3.36) over the interval [t,t+1], where −4≤t≤−1. For all t0≤min{T(R,ω),−4}, by using (3.37) and (4.5), we obtain that
d∫t+1t(||∇p(τ,ω;t0,g0)||2+||∇q(τ,ω;t0,g0)||2)dτ≤α(||p(t)||2+||q(t)||2)+ακ∫t+1t(||∇V(τ)||2+||∇Z(τ)||2)dτ+α∫t+1t2ρz(θτω)(||p(τ)||2+||q(τ)||2)dτ≤αmax−4≤t≤−1C3(t,ω)+ακK1(ω)2d2+2αmax−4≤t≤−1C3(t,ω)∫0−4c|z(θτω)|dτ. | (4.10) |
Therefore, we have
∫t+1t(||∇p(τ,ω;t0,g0)||2+||∇q(τ,ω;t0,g0)||2)dτ≤K4(ω)d, | (4.11) |
where
K4(ω)=αmax−4≤t≤−1C3(t,ω)+ακK1(ω)2d2+2αmax−4≤t≤−1C3(t,ω)∫0−4c|z(θτω)|dτ. |
In this way, for any t0≤min{T(R,ω),−4} and −4≤t≤−1, we deduce that
∫t+1t(||∇U(τ,ω)−∇W(τ,ω)||2+||∇Φ(τ,ω)−∇Ψ(τ,ω)||2)dτ=∫t+1t(||∇p(τ,ω)−∇V(τ,ω)+∇Z(τ,ω)||2+||∇q(τ,ω)||2)dτ≤∫t+1t(2||∇p(τ,ω)||2+||∇q(τ,ω)||2+4(||∇V(τ,ω)||2+||∇Z(τ,ω)||2))dτ≤2K4(ω)d+2K1(ω)d2=K5(ω). | (4.12) |
Thus, for t0≤min{T(R,ω),−4} and −4≤t≤−1, we get by applying (4.9) and (4.12)
∫t+1t(||∇U||2+||∇W||2+||∇Φ||2+||∇Ψ||2)dτ≤14∫t+1t(||(∇U+∇W)+(∇U−∇W)||2+||(∇U+∇W)−(∇U−∇W)||2)dτ+14∫t+1t(||(∇Φ+∇Ψ)+(∇Φ−∇Ψ)||2+||(∇Φ+∇Ψ)−(∇Φ−∇Ψ)||2)dτ≤∫t+1t(||∇U+∇W||2+||∇U−∇W||2+||∇Φ+∇Ψ||2+||∇Φ−∇Ψ||2)dτ=K4(ω)+K5(ω)=K6(ω). | (4.13) |
Step 2. We take the inner product of (2.13) with −ΔU and (2.16) with −ΔW, which follows that
12ddt(||∇U||2+||∇W||2)+d1(||ΔU||2+||ΔW||2)+(b+k)(||∇U||2+||∇W||2)=∫O[−ae−ρz(θtω)(ΔU+ΔW)−e2ρz(θtω)(U2VΔU+W2ZΔW)]dx−∫O[N(ΦΔU+ΨΔW)+D1(|∇U|2−2∇U⋅∇W+|∇W|2)]dx+ρz(θtω)(||∇U||2+||∇W||2)≤34d1(||ΔU||2+||ΔW||2)+a2|O|d1e−2ρz(θtω)+1d1e4ρz(θtω)∫O(U4V2+W4Z2)dx+N2d1(||Φ||2+||Ψ||2)+ρz(θtω)(||∇U||2+||∇W||2). | (4.14) |
Then, by applying (2.2) and (2.4), we have
ddt(||∇U||2+||∇W||2)+2(b+k)(||∇U||2+||∇W||2)≤2a2|O|d1e−2ρz(θtω)+2d1e4ρz(θtω)(||U||4L6||V||2L6+||W||4L6||Z||2L6)+2N2d1(||Φ||2+||Ψ||2)+2ρz(θtω)(||∇U||2+||∇W||2)≤2a2|O|d1e−2ρz(θtω)+2N2d1(||Φ||2+||Ψ||2)+2ρz(θtω)(||∇U||2+||∇W||2)+4δ4d1e4ρz(θtω)[(||U||4+||∇U||4)||V||2L6+(||W||4+||∇W||4)||Z||2L6]≤2a2|O|d1e−2ρz(θtω)+2N2d1(||Φ||2+||Ψ||2)+2ρz(θtω)(||∇U||2+||∇W||2)+4δ4d1e4ρz(θtω)(||U||2+||W||2)2(||V||2L6+||Z||2L6)+4δ4d1e4ρz(θtω)(||∇U||2+||∇W||2)2(||V||2L6+||Z||2L6)≤2a2|O|d1e−2ρz(θtω)+2N2d1(||Φ||2+||Ψ||2)+2ρz(θtω)(||∇U||2+||∇W||2)+(4δ4d1γ2+4δ4d1)e4ρz(θtω)(||∇U||2+||∇W||2)2(||V||2L6+||Z||2L6). | (4.15) |
We can rewrite (4.15) as the following form
dβ1(t)dt≤α1(t)β1(t)+γ1(t),t∈[−3,0], | (4.16) |
where
α1(t)=(4δ4d1γ2+4δ4d1)e4ρz(θtω)(||∇U||2+||∇W||2)(||V||2L6+||Z||2L6)+2ρz(θtω),β1(t)=||∇U||2+||∇W||2,γ1(t)=2a2|O|d1e−2ρz(θtω)+2N2d1(||Φ||2+||Ψ||2). |
For t0≤T(R,ω) and −3≤t≤−1, by applying (4.6) and (4.13), we have
∫t+1tα1(τ)dτ=∫t+1t2ρz(θτω)dτ+∫t+1t(4δ4d1γ2+4δ4d1)e4ρz(θτω)(||∇U||2+||∇W||2)(||V||2L6+||Z||2L6)dτ≤2∫0−3c|z(θτω)|dτ+(4δ4d1γ2+4δ4d1)max−3≤τ≤0[D13(τ,K12d1,ω)e4ρz(θτω)]K6(ω). | (4.17) |
Then, by using (4.13), we obtain
∫t+1tβ1(τ)dτ=∫t+1t(||∇U||2+||∇W||2)dτ≤K6(ω), | (4.18) |
and
∫t+1tγ1(τ)dτ=∫t+1t[2a2|O|d1e−2ρz(θτω)+2N2d1(||Φ||2+||Ψ||2)]dτ≤2a2|O|d1∫0−3e−2ρz(θτω)dτ+2N2d1γK6(ω). | (4.19) |
By applying the uniform Gronwall inequality, which follows that
||U(t,ω;t0,g0)||2E+||W(t,ω;t0,g0)||2E≤K7(ω),t∈[−2,0],t0≤T(R,ω), | (4.20) |
where
K7(ω)=(K6(ω)+2a2|O|d1∫0−3e−2ρz(θτω)dτ+2N2d1γK6(ω))⋅exp[(4δ4d1γ2+4δ4d1)max−3≤τ≤0[D13(τ,K12d1,ω)e4ρz(θτω)]K6(ω)+2∫0−3c|z(θτω)|dτ]. |
Step 3. Taking the scalar product of (2.14) with −ΔV and (2.17) with −ΔZ, and adding up their results, we get
12ddt(||∇V||2+||∇Z||2)+d2(||ΔV||2+||ΔZ||2)=∫O[−bUΔV−bWΔZ+e2ρz(θtω)(U2VΔV+W2ZΔZ)]dx−∫OD2(|∇Z|2−2∇Z⋅∇V+|∇V|2)dx+ρz(θtω)(||∇V||2+||∇Z||2)≤(d24+d24)(||ΔV||2+||ΔZ||2)+b2d2(||U||2+||W||2)+1d2e4ρz(θtω)∫O(U4V2+W4Z2)dx+ρz(θtω)(||∇V||2+||∇Z||2). | (4.21) |
Therefore, we obtain that
ddt(||∇V||2+||∇Z||2)+d2(||ΔV||2+||ΔZ||2)≤2b2d2(||U||2+||W||2)+2d2e4ρz(θtω)(||U||4L6||V||2L6+||W||4L6||Z||2L6)+2ρz(θtω)(||∇V||2+||∇Z||2)≤4δ6d2e4ρz(θtω)[(||U||4+||∇U||4)||∇V||2+(||W||4+||∇W||4)||∇Z||2]+2b2d2(||U||2+||W||2)+2ρz(θtω)(||∇V||2+||∇Z||2)≤4δ6d2e4ρz(θtω)[(||U||2+||W||2)2+(||∇U||2+||∇W||2)2](||∇V||2+||∇Z||2)+2b2d2(||U||2+||W||2)+2ρz(θtω)(||∇V||2+||∇Z||2). | (4.22) |
Then, we can rewrite (4.22) as the following form
dβ2(t)dt≤α2(t)β2(t)+γ2(t),t∈[−2,0], | (4.23) |
where
α2(t)=4δ6d2e4ρz(θtω)[(||U||2+||W||2)2+(||∇U||2+||∇W||2)2]+2ρz(θtω),β2(t)=||∇V||2+||∇Z||2,γ2(t)=2b2d2(||U||2+||W||2). |
For t0≤T(R,ω) and −2≤t≤−1, by applying (4.20), we infer that
∫t+1tα2(τ)dτ=∫t+1t4δ6d2e4ρz(θτω)[(||U||2+||W||2)2+(||∇U||2+||∇W||2)2]dτ+∫t+1t2ρz(θτω)dτ≤(4δ6d2γ2+4δ6d2)max−2≤τ≤0e4ρz(θτω)K27(ω)+2∫0−2c|z(θτω)|dτ. | (4.24) |
Then, by using (4.3), we obtain
∫t+1tβ2(τ)dτ=∫t+1t(||∇V||2+||∇Z||2)dτ≤K1(ω)2d2, | (4.25) |
and by applying (4.20), we have
∫t+1tγ2(τ)dτ=∫t+1t2b2d2(||U||2+||W||2)dτ≤2b2d2γK7(ω). | (4.26) |
Owing to the uniform Gronwall inequality, which follows that
||V(t,ω;t0,g0)||2E+||Z(t,ω;t0,g0)||2E≤K8(ω),t∈[−1,0],t0≤T(R,ω), | (4.27) |
where
K8(ω)=(K1(ω)2d2+2b2d2γK7(ω))⋅exp[(4δ6d2γ2+4δ6d2)max−2≤τ≤0e4ρz(θτω)K27(ω)+2∫0−2c|z(θτω)|dτ]. |
Step 4. Taking the scalar product of (2.15) with −ΔΦ and (2.18) with −ΔΨ, and adding up their results, we get
12ddt(||∇Φ||2+||∇Ψ||2)+d3(||ΔΦ||2+||ΔΨ||2)+(λ+N)(||∇Φ||2+||∇Ψ||2)=∫O(−kUΔΦ−kWΔΨ)dx−∫OD3(|∇Φ|2−2∇Φ⋅∇Ψ+|∇Ψ|2)dx+ρz(θtω)(||∇Φ||2+||∇Ψ||2)≤d32(||ΔΦ||2+||ΔΨ||2)+k22d3(||U||2+||W||2)+ρz(θtω)(||∇Φ||2+||∇Ψ||2). | (4.28) |
Then, we have
ddt(||∇Φ||2+||∇Ψ||2)+d3(||ΔΦ||2+||ΔΨ||2)≤k2d3(||U||2+||W||2)+2ρz(θtω)(||∇Φ||2+||∇Ψ||2). | (4.29) |
Then, we can rewrite (4.29) as the following form
dβ3(t)dt≤α3(t)β3(t)+γ3(t),t∈[−2,0], | (4.30) |
where
α3(t)=2ρz(θtω),β3(t)=||∇Φ||2+||∇Ψ||2,γ3(t)=k2d3(||U||2+||W||2). |
For t0≤T(R,ω) and −2≤t≤−1, we infer that
∫t+1tα3(τ)dτ=∫t+1t2ρz(θτω)dτ≤2∫0−2c|z(θτω)|dτ, | (4.31) |
then, by using (4.13), we obtain
∫t+1tβ3(τ)dτ=∫t+1t(||∇Φ||2+||∇Ψ||2)dτ≤K6(ω), | (4.32) |
and applying (4.20), we have
∫t+1tγ3(τ)dτ=∫t+1tk2d3(||U||2+||W||2)dτ≤k2d3γK7(ω). | (4.33) |
Owing to the uniform Gronwall inequality, which follows that
||Φ(t,ω;t0,g0)||2E+||Ψ(t,ω;t0,g0)||2E≤K9(ω),t∈[−1,0],t0≤T(R,ω), | (4.34) |
where
K9(ω)=(K6(ω)+k2d3γK7(ω))⋅exp{2∫0−2c|z(θτω)|}. |
Finally, let t=0 in (4.20), (4.27) and (4.34). Therefore, (4.1) holds with K(ω)=K7(ω)+K8(ω)+K9(ω). In this way, we have completed the proof of Lemma 4.2.
In this section, we obtain the existence of a pullback random attractor for the stochastic extended Brusselator system in H.
Theorem 5.1. The extended Brusselator random dynamical system f has a unique pullback random attractor.
Proof. In Lemma 3.2, we obtained that the RDS f has a bounded pullback absorbing set associated with the universe D. In Lemma 4.2, due to the embedding E↪H is a compact mapping, which means that the RDS f is pullback asymptotically compact in H by applying the results of Proposition 1.8.3 in [28]. Then, by Proposition 2.1, the existence of pullback random attractor A(ω)={A(ω)}ω∈Ω is proved for the RDS f, which is given by
A(ω)=⋂τ≥0¯⋃t≥τf(t,θ−tω,B0(θ−tω)). | (5.1) |
Therefore, we have completed the proof of Theorem 5.1.
In this paper, we prove the existence of random attractors for stochastic dynamics by using the exponential transformation of the Ornstein-Uhlenbeck process to replace the exponential transformation of Brownian motion, which changes the structure of the original Brusselator equations and produces the non-autonomous terms, cf. (2.13)–(2.18). Based on this, we have to overcome the difficulties of coupling structure and make more complex estimates.
If ρ=0, system (1.1)–(1.6) reduces to the extended Brusselator system without random terms, which has been established the global dissipative dynamics by You and Zhou in [15]. Furthermore, Tu and You [16] proved random attractor of stochastic Brusselator system with a multiplicative noise, this paper has included the results of [16] when w,z,φ,ψ=0.
The authors thank the referees for their careful reading of the manuscript and insightful comments, which helped to improve the quality of the paper. We would also like to acknowledge the valuable comments and suggestions from the editors, which vastly contributed to improving the presentation of the paper. This work was supported by the NSF of China (Nos. 11371183, 11901342), the NSF of Shandong Province (Nos. ZR2018QA002, ZR2019MA067) and the China Postdoctoral Science Foundation (No. 2019M652350).
The authors declare that they have no conflicts of interest.
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