# Random Attractors for Stochastic Three-Component Reversible Gray-Scott System with Multiplicative White Noise

Authors: Anhui Gu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The paper is devoted to proving the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with multiplicative white noise.


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## Body

## 1.  Introduction

Let𝒪 be an open bounded set of Rn(n≤3) with a locally Lipschitz continuous boundary ∂𝒪. We consider the stochastic three-component reversible Gray-Scott system with multiplicative noise(1.1)∂ũ∂t=d1▵ũ-(F+k)ũ+ũ2ṽ-Gũ3+Nw̃+σũ∘dBtdt,∂ṽ∂t=d2▵ṽ+F(1-ṽ)-ũ2ṽ+Gũ3+σṽ∘dBtdt,∂w̃∂t=d3▵w̃+kũ-(F+N)w̃+σw̃∘dBtdt,
where ũ=ũ(x,t), ṽ=ṽ(x,t), and w̃=w̃(x,t) are real-valued functions on 𝒪×[τ,+∞), τ∈R; all the parameters are arbitrarily given positive constants; Bt is a Brownian motion and ∘ denotes the Stratonovich sense of the stochastic term. In this work, we consider the homogenous Neumann boundary condition(1.2)∂ũ∂ν(x,t)=∂ṽ∂ν(x,t)=∂w̃∂ν(x,t)=0,x∈∂O,
where ∂/∂ν is the outward normal derivative, and with an initial condition(1.3)ũ(x,0)=ũ0(x),ṽ(x,0)=ṽ0(x),w̃(x,0)=w̃0(x),x∈O.The three-component reversible Gray-Scott model was firstly introduced by Mahara et al. [1]. Recently in [2], You gave the existence of global attractor for system (1.1) when σ=0 with Neumann boundary condition (1.2) on a bounded domain of space dimension n≤3 by the method of the rescaling and grouping estimate. However, the reactions and diffusions are often affected by stochastic factors then it is important and meaningful to take the asymptotic behavior of solutions to consideration. Particularly, the dynamics of certain systems frequently follows some self-organization process where the development of new, complex structures takes place primarily in and through the system itself. This self-organization is normally triggered by internal variation processes, which are usually called fluctuations or noise, that have a positive influence on the system. For instance, recent theoretical studies and experiments with cultured glial cells and the Belousov-Zhabotinsky reaction have shown that noise may play a constructive role on the dynamical behavior of spatially extended systems [3–5]. Therefore, one cannot ignore the role of noise in chemical and biological self-organization and its relationship with the environmental selection of emergent patterns [6]. In [7–9], the influence of additive noise on Gray-Scott systems was discussed. As pointed in [10, 11], the effects of additive and multiplicative noises are fundamentally different in nonlinear systems. While the effect of additive noise does not depend on the state of the system, the effect of multiplicative noise is state dependent. Natural systems in which the effect of noise on the system's dynamics does depend on the recent state are autocatalytic chemical reactions or growth processes in developmental biology. More generally speaking, in each system whose dynamics shows some degree of self-referentiality, the effect of exogenous noise will depend on the recent system's state. If noise is multiplicative, “new” phenomena can occur; that is, the noisy system can exhibit behavior, which is qualitatively different from that of the deterministic system, a phenomenon that has been coined noise-induced transitions.A fundamental problem in the study of dynamics of a stochastic partial differential equation is to show that it generates a random dynamical system (or stochastic flow). One of the most interesting concepts of the theory of random dynamical systems is the random attractor, which was introduced in the 90s of the last century (see [12]). An attractor for an autonomous dynamical system is a compact set in the phase space, attracting the image of particular sets of initial states under the evolution of the dynamical system. However, the random case is more complicated, because random attractors depend on the random parameter and have their own temporal dynamics induced by the noise (cf. the definition in Section 3). Moreover, the existence of a random attractor to the stochastic reversible Gray-Scott system, especially of three components, is widely open to the best of our knowledge. According to methodology of [2] of nondissipative coupling of three variables and the coefficients barrier, we consider system (1.1)–(1.3), which gives partly an answer to the problems of random perturbations proposed in [13]. In this paper, we use the notions and frameworks in [12, 14, 15] to study the stochastic three-component reversible Gray-Scott system with multiplicative white noise.The paper is organized as follows. In Section2, we give the existence and uniqueness of solution. Section 3 is devoted to the existence of a random attractor.


## 2.  Existence and Uniqueness of Solutions

Let(Ω,ℱ,P) be a probability space, and {θt:Ω→Ω,t∈R} is a family of measure preserving transformations such that (t,ω)↦θtω is measurable, θ0=id, and θt+s=θtθs for all s,t∈R. The flow θt together with the probability space (Ω,ℱ,P,(θt)t∈R) is called as a measurable dynamical system.A random dynamical system (RDS) on a Polish space(X,d) with Borel σ-algebra ℬ over θ on (Ω,ℱ,P) is a measurable map(2.1)φ:R+×Ω×X⟶X,(t,ω,x)⟼φ(t,ω)x,
such that P-almost surely (P-a.s.) we have(i)
φ(0,ω)=id on X;(ii)
(cocycle property)φ(t+s,ω)=φ(t,θsω)φ(s,ω) for all s,t≥0.An RDS is continuous or differentiable ifφ(t,ω):X→X is continuous or differentiable.A mapB:Ω→2X is said to be a closed (compact) random set if B(ω) is closed (compact) for P-a.s. ω∈Ω and if ω↦d(x,B(ω)) is P-a.s. measurable for all x∈X.Consider the product Hilbert spacesL2(𝒪), H=[L2(𝒪)]3, and H1(𝒪), E=[H1(𝒪)]3 with the usual inner products and norms(2.2)(u,v)=∫Ωuvdx,|u|=(u,u)1/2∀u,v∈L2(O),((u,v))=∑i=1n(Diu,Div)+F(u,v),‖u‖=((u,u))1/2∀u,v∈H1(O).
Obviously, for fixed F the scalar product and norm defined above are equivalent to the usual scalar product and norm in H1(𝒪). And the norm of Lp(𝒪) will be denoted by |·|Lp if p≠2,X=(L6(𝒪))3.Define the unbounded positive linear operator(2.3)A(D(A)⟶H)=Δ̃-FI,
where(2.4)Δ̃=(d1▵000d2▵000d3▵),D(A)={(ϕ,φ,ψ)∈[H2(O)]3:∂ϕ∂ν=∂φ∂ν=∂ψ∂ν=0on⁡∂O}.
By the Lumer-Phillips theorem and the generation theorem for analytic semigroup [16], the operator A in (2.3) is linear, sectorial, closed, and defined and is the generator of an analytic C0-semigroup {eAt,t≥0} on the Hilbert space H. Its spectral set consists of only nonnegative eigenvalues, denoted by μi=λi+F, i≥0, where λi are the corresponding eigenvalues of Δ̃ satisfying(2.5)0=λ0<λ1≤λ2≤⋯≤λi≤⋯(λi⟶∞asi⟶∞).
By the fact that H1(𝒪)↪L6(𝒪) is a continuous embedding for n≤3 and by the generalized Hölder inequality, one has(2.6)|u2v|≤|u|L62|v|L6,foru,v∈L6(O).
Therefore, the nonlinear mapping defined on E,(2.7)f(ũ,ṽ,w̃)=(-kũ+ũ2ṽ-Gũ3+Nw̃F-ũ2ṽ+Gũ3kũ-Nw̃):E⟶H,
is locally Lipschitz continuous. Thus, the initial boundary problem (1.1)–(1.3) is formulated as an initial value problem of the stochastic three-component reversible Gray-Scott system with multiplicative noise(2.8)dg̃dt=Ag̃+f(g̃)+σg̃∘dBtdt,
and an initial condition(2.9)g̃(0)=g̃0=(ũ0,ṽ0,w̃0)∈H,
where g̃(t)=(ũ(t,·),ṽ(t,·),w̃(t,·)). Bt is a one-dimensional two-sided Wiener process on a probability space (Ω,ℱ,P), where(2.10)Ω={ω∈C(R,Rm):ω(0)=0},
the Borel σ-algebra ℱ on Ω is generated by the compact open topology, and P is the corresponding Wiener measure on ℱ. We can define a family of measure-preserving and ergodic transformations (a flow) {θt}t∈R by(2.11)θtω(⋅)=ω(⋅+t)-ω(t).
By means of the change of variables(2.12)u(t)=α(t)ũ(t),v(t)=α(t)ṽ(t)w(t)=α(t)w̃(t),withα(t)=e-σBt,
system (1.1) can be written as
(2.13)∂u∂t=d1▵u-(F+k)u+α-2(t)u2v-Gα-2(t)u3+Nw,(2.14)∂v∂t=d2▵v+F(α(t)-v)-α-2(t)u2v+Gα-2(t)u3,(2.15)∂w∂t=d3▵w+ku-(F+N)w.
That is g(t,·)=(u(t,·),v(t,·),w(t,·)) satisfies(2.16)dgdt=Ag+f̃(g,ω),
with initial condition(2.17)g(0)=g̃(0)=g0=(u0,v0,w0)∈H,
where(2.18)f̃(g,ω)=(-ku+α-2(t)u2v-Gα-2(t)u3+NwF-α-2(t)u2v+Gα-2(t)u3ku-Nw).
Due to the fact that H1(𝒪)↪L6(𝒪) and (2.6), we know that f̃(g,ω) is locally Lipschitz continuous with respect to g and bounded for every ω∈Ω. By the same method in [17, Chapters II and XV], we can prove for P-a.s. every ω∈Ω the local existence and uniqueness of the weak solution g(t), t∈[τ,T] for some τ<T, of (2.16) with g(τ)=g0, which is continuously depending on the initial data g0∈H and turns out to be a strong solution on (τ,T] by [16, Theorem 48.5]. One can show that for P-a.s. every ω∈Ω, the following statements hold for all τ<T.(i)
Ifg̃(0,ω)∈H, then g̃(t,ω) lies in
(2.19)C([τ,T);H)∩C1((τ,T);H)∩L2([τ,T);E).(ii)
g̃(t,g̃(0,ω)) is jointly continuous in t and g̃(0,ω) in [τ,T)×H.(iii)
The solution mapping of (2.16) satisfies the property of an RDS.This system has a unique solution for everyω∈Ω. Hence the solution mapping(2.20)S̃(t,ω):g̃(τ,ω)⟼g̃(t,ω)
generates an RDS. So the transformation(2.21)Ŝ(t,ω):α-1(t)g̃(τ,ω)⟼α-1(t)g̃(t,ω)
also determines an RDS corresponding to system (1.1).We will prove the existence of a nonempty compact random attractor for the RDSŜ(t,ω).


## 3. Existence of a Random Attractor

A random set𝒦(ω) is said to absorb the set B⊂X for an RDS φ if P-a.s. there exists tB(ω) such that(3.1)φ(t,θ-tω)B⊂K(ω)∀t≥tB(ω).
A random set 𝒜(ω) is said to be a random attractor associated to the RDS φ if P-a.s.:(i)
𝒜(ω) is a random compact set, that is, P-a.s. ω∈Ω, 𝒜(ω) is compact, and for all x∈X and P-a.s. the map x↦dist(x,𝒜(ω)) is measurable.(ii)
φ(t,ω)𝒜(ω)=A(θtω) for all t≥0 (invariance).(iii)
For all boundedB⊂X,
(3.2)lim⁡t→∞dist⁡(φ(t,θ-tω)B,A(ω))=0,
where dist(·,·) denotes the Hausdorff semidistance:
(3.3)dist⁡(Y,Z)=sup⁡inf⁡x∈Yy∈Zd(x,y),Y,Z∈X.Proposition 3.1 (see [14, 15]).
Letϕ be an RDS on a Polish space  (X,d) with Borel  σ-algebra  ℬ over the flow  {θt}t∈R on a probability space  (Ω,ℱ,P). Suppose there exists a random compact set  𝒦(ω) such that for any bounded nonrandom set  B⊂X  P-a.s
(3.4)dist⁡(φ(t,θ-tω)B,K(ω))⟶0ast⟶+∞.
Then the set
(3.5)A(ω)=⋃B⊂XΛB(ω)¯
is a unique random attractor for ϕ, where the union is taken over all bounded B⊂X and ΛB(ω) is the omega-limit set of B given by
(3.6)ΛB(ω)=⋂s≥0⋃t≥sϕ(t,θ-tω)B¯.Now, we will show the existence of a random attractor for the RDS (2.16).Lemma 3.2.
There exists a random variabler1(ω)>0, depending on  F,  G,  σ, and  μ, such that for all  ρ>0 there exists  t(ω)≤-1 such that the following holds  P-a.s. For all  t0≤t(ω), and for all  g̃0∈H with  |g̃0|≤ρ, the solution  g(t,ω;t0,α(t0)g̃0) of system (2.16) over  [t0,∞), with  g(t0)=α(t0,ω)g̃0, satisfies the inequality
(3.7)|g(-1,ω;t0,α(t0,ω)g̃0)|2≤r12(ω).Proof.
Define(3.8)W(t,x)=Nkw(t,x),μ=kN.
Then (2.13)–(2.15) can be written as
(3.9)∂u∂t=d1▵u-(F+k)u+α-2(t)u2v-Gα-2(t)u3+kW,(3.10)∂v∂t=d2▵v+F(α(t)-v)-α-2(t)u2v+Gα-2(t)u3,(3.11)μ∂W∂t=μd3▵W+ku-(μF+k)W.
Taking the inner products of (3.9)–(3.11) with Gu(t), v(t), and GW(t), respectively. Then sum up the resulting equalities. By the Neumann boundary condition (1.2), we get
(3.12)12ddt(G|u|2+|v|2+μG|W|2)+G(F+k)|u|2+F|v|2+G(μF+k)|W|2+d1G‖u‖2+d2‖v‖2+μGd3‖W‖2≤∫ΩFα(t)vdx+2kG∫ΩuWdx-α-2(t)∫Ω(Gu2-uv)2dx≤kG|u|2+kG|W|2+F2|v|2+Fα2(t)2|O|,
where |𝒪| denotes the volume of 𝒪. Set
(3.13)d=min⁡{d1,d2,d3},C1=F|O|min⁡{1,G,G/μ}.
Then (3.12) yields
(3.14)ddt|g(t)|2+d‖g(t)‖2+F|g(t)|2≤C1α2(t).
Applying Gronwall's inequality to (3.14) and then integrating in [t0,-1], with t0≤-1 we have
(3.15)|g(-1)|2≤|α(t0)g0|2e-F(-1-t0)+C1∫t0-1e-F(-1-s)α2(s)ds≤eF(eFt0|α(t0)g0|2+C1∫t0-1eFsα2(s)ds).
Consequently, give B(0,ρ)⊂H, P-a.s. there exists t(ω,ρ)≤-1 such that for t0≤t(ω,ρ) and all g0∈B(0,ρ),
(3.16)|g(-1,ω;t0,α(t0)g̃0)|2≤r12(ω),
with
(3.17)r12(ω)=eF(1+C1∫-∞-1eFsα2(s)ds).
Indeed, it is enough to choose t(ω,ρ) such that
(3.18)eFt0α2(t0)ρ2≤1,
and take into account (3.15) and the fact that P-a.s. eFsα2(s)=eFse-2σWs→0 as s→-∞.If we now return to (3.14) and integrate for t∈[-1,0], we have(3.19)|g(t)|2≤|g(-1)|2e-F(t+1)+C1∫-1te-F(t-s)α2(s)ds,d∫-10‖g(s)‖2ds≤|g(-1)|2+C1∫-10α2(s)ds.
Thus, we can conclude that given B(0,ρ)⊂H and P-a.s. there exists t(ω,ρ)≤-1 such that for t0≤t(ω,ρ) and for all g0∈B(0,ρ),(3.20)|g(t,ω;t0,g0)|2≤e-F(t+1)r12(ω)+C1∫-1te-F(t-s)α2(s)ds,d∫-10‖g(s,ω;t0,g0)‖2ds≤r12(ω)+C1∫-10α2(s)ds.To prove the absorption at timet=0, we need the following proposition.Proposition 3.3.
There exists a random variabler3(ω)>0, depending on  λ1,  σ, and  d, such that for all  ρ′>0 there exists  t(ω)≤-1 such that the following holds P-a.s. For all  t0≤t(ω) and for all  g̃0∈H with  |g̃0|≤ρ′, the solution  g(t,ω;t0,α(t0)g0) of system (2.16) over  [t0,∞), with  g(t0)=α(t0,ω)g̃0, satisfies the inequality
(3.21)|g(-1,ω;t0,α(t0,ω)g̃0)|X6≤r32(ω).
Also, for t∈[-1,0] one has
(3.22)|g(t)|X6≤e-F(t+1)r32(ω)+C4∫-1te-F(t-τ)α6(τ)dτ.Proof.
LettingV(t,x)=v(t,x)/G, (3.9)–(3.11) can be written as
(3.23)∂u∂t=d1▵u-(F+k)u+Gα-2(t)u2V-Gα-2(t)u3+kW,(3.24)∂V∂t=d2▵v+FGα(t)-FV-α-2(t)u2V+α-2(t)u3,(3.25)μ∂W∂t=μd3▵W+ku-(μF+k)W.
Take the inner products ((3.23), u5(t)), ((3.24), GV5(t)) and ((3.25), W5(t)) and sum up the resulting equalities. By the Neumann boundary condition, we get
(3.26)16ddt(|u|L66+G|V|L66+μ|W|L66)+5(d1‖u2∇u‖2+d2G‖V2∇V‖2+μd3G‖W2∇W‖2)=-(F+k)∫Ωu6ds+F∫Ωα(t)V5dx-GF∫ΩV6dx-(μF+k)∫ΩW6dx+k∫Ωu5Wdx+k∫ΩuW5dx-Gα-2(t)∫Ω(u8-u7V-u3V5+u2V6)dx.
By using Young's inequality, we obtain
(3.27)-Gα-2(t)∫Ω(u8-u7V-u3V5+u2V6)dx≤0,k∫Ωu5Wdx+k∫ΩuW5dx≤k|u|L66+k|W|L66.
From (3.27), (3.26) yields
(3.28)ddt(|u|L66+G|V|L66+μ|W|L66)+F(|u|L66+G|V|L66+μ|W|L66)≤F|O|G5α6(t),
that is,
(3.29)ddt(|u|L66+G-5|v|L66+μ-5|w|L66)+F(|u|L66+G-5|v|L66+μ-5|w|L66)≤F|O|G5α6(t).
By denoting
(3.30)C4=F|O|G5min⁡{1,G-5,μ-5},
then (3.29) implies that
(3.31)ddt|g(t)|X6+F|g(t)|X6≤C4α6(t).
Integrating in [t0,-1], with t0≤-1 we have
(3.32)|g(-1)|X6≤|α(t0)g0|X6e-F(-1-t0)+C4∫t0-1e-F(-1-s)α6(s)ds.
Consequently, given B(0,ρ′)⊂H, P-a.s. there exists t(ω,ρ′)≤-1 such that for all t0≤t(ω,ρ′) and for all g0∈B(0,ρ′),
(3.33)|g(-1,ω,t0,α(t0)g0)|X6≤r32(ω),
with
(3.34)r32(ω)=eF(1+C4∫-∞-1eFsα6(s)ds).
In fact, it is enough to choose t(ω,ρ′) to satisfy
(3.35)eFt0α6(t0)ρ′6≤1,
and take into (3.32) and the fact that P-a.s. eFsα6(s)=eFse-6σWs→0 as s→-∞. Also, from (3.32) and for t∈[-1,0] we get
(3.36)|g(t)|X6≤e-F(t+1)r32(ω)+C4∫-1te-F(t-τ)α6(τ)dτ.Lemma 3.4.
There exists a random variabler2(ω)>0, depending on  F,  G,  N,  d,  k, and  σ, such that for all  ρ>0 there exists  t(ω)≤-1 such that the following holds P-a.s. For all  t0≤t(ω) and for all  g̃0∈H with  |g̃0|≤ρ, there exists a unique solution  g(t,ω;t0,α(t0)g0) of system  (2.16) over  [t0,∞), with  g(t0)=α(t0,ω)g̃0, and put  g̃(t,ω;t0,g̃0):=α-1(t,ω)g(t,ω;t0,g̃0). Then
(3.37)‖g̃(0,ω;t0,g̃0)‖2≤r22(ω).Proof.
To get a bound inE, we multiply (2.13)–(2.15) by -Δu, -Δv, and –Δw, respectively. Add up the three equalities, and due to the Neumann boundary condition, we have
(3.38)12ddt(‖u‖2+‖v‖2+‖w‖2)+d1|▵u|2+d2|▵v|2+d3|▵w|2+(F+k)‖u‖2+F‖v‖2+(F+N)‖w‖2=-α-2(t)∫Ωu2v▵udx+Gα-2(t)∫Ωu3▵udx-N∫Ωw▵udx-Fα(t)∫Ω▵vdx+α-2(t)∫Ωu2v▵vdx-Gα-2(t)∫Ωu3▵vdx-k∫Ωu▵wdx≤d12|▵u|2+d22|▵v|2+d32|▵w|2+α-4(t)(1d1+1d2)∫Ωu4v2dx+N2d1∫Ωw2dx+k22d3∫Ωu2dx+G2α-4(t)d2∫Ωu6dx,
that is,
(3.39)ddt(‖u‖2+‖v‖2+‖w‖2)+F(‖u‖2+‖v‖2+‖w‖2)≤α-4(t)(43d1+43d2+2G2d2)∫Ω(u6+v6)dx+2N2d1|w|2+k2d3|u|2≤C2α-4(t)|g|X6+C3|g|2.
Here d=min{d1,d2,d3}, C2=(1/d)(8/3+2G2), C3=(1/d)max{2N2,k2}. Then from (3.39) we have
(3.40)ddt‖g‖2+F‖g‖2≤C2α-4(t)|g|X6+C3|g|2.
Integrating (3.40) in [s,0], s∈[-1,0], we obtain
(3.41)‖g(0)‖2≤‖g(s)‖2+C2∫s0α-4(τ)|g(τ)|X6dτ+C3∫s0|g(τ)|2dτ.
Integrating (3.41) in [-1,0] and by (3.20),
(3.42)‖g(0)‖2≤∫-10‖g(τ)‖2dτ+C2∫-10α-4(τ)|g(τ)|X6dτ+C3∫-10|g(τ)|2dτ≤(1d+C3F)(r12(ω)+C1∫-10α2(τ)dτ)+C2sup⁡-1≤t≤0α-4(t)∫-10|g(τ)|X6.
It is now straightforward from (3.31) and (3.36) that
(3.43)‖g(0)‖2=‖g̃(0)‖2≤(1d+C3F)(r12(ω)+C1∫-10α2(τ)dτ)+C2Fsup⁡-1≤t≤0α-4(t)(r32(ω)+C4∫-10α6(τ)dτ).
Consequently, P-a.s. there exists r2(ω) such that given ρ>0, there exists t̃(ω)≤-1 such that for all t0≤t̃(ω) and g̃0∈H with |g̃0|≤ρ,
(3.44)‖g̃(0,ω;t0,g̃0)‖2≤r22(ω),
where
(3.45)r22(ω)=(C5+C6)eF+C1C5∫-∞-1eF(s+1)α2(s)ds+C1C5∫-10α2(s)ds+C4C6∫-∞-1eF(s+1)α6(s)ds+C4C6∫-10α6(s)ds,C5=(1d+C3F),C6=C2Fsup⁡-1≤t≤0α-4(t).Thus, we can have the main result.Theorem 3.5.
The RDSŜ(t,ω) has a nonempty compact random attractor  𝒜(ω).Proof.
This follows from Lemma3.2 and Lemma 3.4 combined with the embedding of E↪H and Proposition 3.1.Remark 3.6.
It is necessary and interesting for us to consider the Hausdorff dimension of the random attractor which is generated by the stochastic three-component reversible Gray-Scott system with multiplicative white noise, but it seems impossibile to apply the results in [18, 19] directly because of the higher-order terms. In order to verify the differentiability properties for the cocycle generated by the random system, we need to check condition (2.7)–(2.13) in [19]. Considering the linearized equation (2.13)–(2.15), we have
(3.46)∂Ũ∂t=d1▵Ũ-(F+k)Ũ+2uve2σWtŨ+u2Ṽe2σWt-3Ge2σWtu2Ũ+NW̃,∂Ṽ∂t=d1▵Ṽ-FṼ-2uve2σWtŨ-u2Ṽe2σWt+3Gu2e2σWtŨ,∂W̃∂t=d1▵W̃+kŨ-(F+N)W̃.
Letting Φ(t)=(Ũ,Ṽ,W̃),q(t)=g̃(t)-g̅(t)-Φ(t), then
(3.47)q(t)=(q1(t),q2(t),q3(t))=(ũ-u̅-Ũ,ṽ-v̅-Ṽ,w̃-w̅-W̃)
satisfies
(3.48)∂q1∂t=d1▵q1-(F+k)q1+α(t)(ũ2ṽ-u̅2v̅)-2ũṽe2σWt(t)Ũ-ũ2e2σWtṼ+3Gũ2e2σWtŨ-Gα(t)(ũ3-u̅3)+Nq3,∂q2∂t=d1▵q2-Fq2-α(t)(ũ2ṽ-u̅2v̅)+2ũṽe2σWtŨ+ũ2e2σWtṼ-3Gũ2e2σWtŨ+Gα(t)(ũ3-u̅3),∂q3∂t=d1▵q3+kq1-(F+N)q3.
Here, let
(3.49)g̃(t)=g̃(t,ω;0,g̃0)=(ũ(t,ω;0,ũ0),ṽ(t,ω;0,ṽ0),w̃(t,ω;0,w̃0)),g̅(t)=g̅(t,ω;0,g̅0)=(u̅(t,ω;0,u̅0),v̅(t,ω;0,v̅0),w̅(t,ω;0,w̅0))
be two solutions of system (1.1). From (3.48), it seems hard to get the conclusion.



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*Source: 810198-2012-05-10.xml*