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Open Mathematics

formerly Central European Journal of Mathematics

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Volume 14, Issue 1 (Jan 2016)

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Random attractors for stochastic two-compartment Gray-Scott equations with a multiplicative noise

Xiaoyao Jia
  • Corresponding author
  • School of Mathematics and Statistics, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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/ Juanjuan Gao
  • School of Mathematics and Statistics, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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/ Xiaoquan Ding
  • School of Mathematics and Statistics, Henan University of Science and Technology, No.263 Kai-Yuan Road, Luo-Long District, Luoyang, Henan Province, 471023, China
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Published Online: 2016-09-08 | DOI: https://doi.org/10.1515/math-2016-0052

Abstract

In this paper, we consider the existence of a pullback attractor for the random dynamical system generated by stochastic two-compartment Gray-Scott equation for a multiplicative noise with the homogeneous Neumann boundary condition on a bounded domain of space dimension n ≤ 3. We first show that the stochastic Gray-Scott equation generates a random dynamical system by transforming this stochastic equation into a random one. We also show that the existence of a random attractor for the stochastic equation follows from the conjugation relation between systems. Then, we prove pullback asymptotical compactness of solutions through the uniform estimate on the solutions. Finally, we obtain the existence of a pullback attractor.

Keywords: Gray-Scott equation; Random attractor; Random dynamical system; Multiplicative noise

MSC 2010: 35K45; 35B40

1 Introduction

In this paper, we consider the following coupled stochastic two-compartment Gray-Scott equation, which is a reaction-diffusion system with multiplicative noise: u~t=d1Δu~F+ku~+u~2v~+D1w~u~+σu~dWtdt,(1) v~t=d2Δv~+F1v~u~2v~+D2y~v~+σv~dWtdt,(2) w~t=d1Δw~F+kw~+w~2y~+D1u~w~+σw~dWtdt,(3) y~t=d2Δy~+F1y~w~2y~+D2v~y~+σy~dWtdt,(4)

for t > 0, on a bounded domain D. Here D is an open bounded set of ℝn (n ≤ 3), and it has a locally Lipschitz coutinuous boundary ∂D. Suppose that the equations have the following homogeneous Neumann boundary condition: u~vt,x=v~vt,x=w~vt,x=y~vt,x=0,t>0,xD,(5)

where v is the outward normal derivative, and have the following initial condition u~0,x=u~0x,v~0,x=v~0x,w~0,x=w~0x,y~0,x=y~x,xD.(6)

Here d1, d2, F, k, D1 and D2 are positive constants; σ is a positive parameter; Δ is the Laplacian operator with respect to xD;u~x,t, u~x,t, u~x,t and y~x,t are real functions of xD; and Wt is a two-sided real-valued Wiener process on a probability space (Ω, F , ℙ). Here Ω=ωCR,R:ω0=0

the Borel sigma-algebra F on Ω is generated by the compact open topology (see [1]), ℙ is the corresponding Wiener measure on F o denotes the Stratonovich sense in the stochastic term. We identify ω(t) with Wt (ω), i.e. Wt (ω) = W(t, ω) = ω(t), t ∈ ℝ.

The Gray-Scott equation is a kind of very important reaction-diffusion system, which arises from many chemical or biological systems [2-5]. This equation has been researched by many authors (see [2-10]). One of the most important problems in mathematical physics is the asymptotic behavior of dynamical system, which has been developed greatly in recent years. For the deterministic system, the global attractor is a very important tool to study the asymptotic behavior of dynamical system (see [9-16]). If σ = 0, system (1)-(4) reduces to the two-compartment Gray-Scott equation without random terms, which has been investigated by You [10], where we proved the existence of the global attractor for the coupled two-compartment Gray-Scott equations with homogeneous Neumann boundary condition on a bounded domain.

Stochastic differential equations of this type arise from many chemical or biological systems when random spatiotemporal force is taken into consideration. These random perturbations play important roles in macroscopic phenomena. To study the properties of stochastic dynamical systems, the concept of pullback random attractor is introduced [1, 17, 18]. The existence of random attractors for stochastic dynamical systems has been studied [6, 7, 19-21]. In this paper, we study the existence of random attractor for stochastic two-compartment Gray-Scott equation on bounded domain D of space dimension n ≤ 3.

The paper is organized as follows. In Section 2, we recall a theorem about the existence of random pullback attractor for random dynamical system, and transform the stochastic system (1)-(6) into a continuous random dynamical system (18)-(19) Ornstein-Uhlenbeck process. Moreover, we show that, for each ω, the random dynamical system has a unique global solution. In Section 3, we obtain some uniform estimates of solutions for system (18)-(19) as t → ∞. These estimates are used to prove the existence of bounded absorbing sets and the asymptotic compactness of the solutions. In the last section, we obtain the existence of a pullback random attractor.

The following notations will be used throughout this paper. ‖ · ‖ and (·, ·) denote the norm and the inner product in L2(D) or [L2(D)]4 respectively. ‖ · ‖Lp and ‖ · ‖H1 are used to denote the norm in Lp(D) and H1(D).

By the Poincaré’s inequality, there is a constant γ > 0 such that ϕ2γϕ2,forϕH01Dor[H01D]4.(7)

Note that H01DL6Dforn3. There exists a constant η > 0 such that the following embedding inequality holds: fH12ηfL62,forfH01Dor[H01(D)]4.(8)

2 RDS Generated by Stochastic Gray-Scott equation

In this section, we first recall a theorem for the existence of random attractors. Please note that here we omit the basic knowledge about random dynamical systems (RDS) and random attractor. Reader can refer to [1, 11, 17-19] for these knowledge.

Suppose that (X, ‖ · ‖X) is a separable Banach space with Borel σ—algebra Β(X), and (Ω, F , ℙ) is a probability space. Let (Ω, F , ℙ, (θt) t∈ℝ) be a metric dynamical system, and assume that ϕ is a continuous RDS on X over (Ω, F , ℙ, (θt) t∈ℝ). We recall a Proposition which will be used to prove the existence of random attractor ([11], [19]) for RDS.

Suppose D is the collection of random subsets of X, and {K(ω)}ω∈Ω, ∈ D is a random absorbing set for RDS ϕ in D and ϕ is D-pullback asymptotically compact in X. Then ϕ has a unique D-random attractor {A(ω)}ω∈Ω which has the following form Aω=τ0tτϕt,θtω,Kθtω.¯(9)

Next, we shall show that system (1)-(6) generates a random dynamical system. For our purpose, we first transform this stochastic system into a deterministic dynamical with random attractor. Assume that (Ω, ℱ, ℙ) is the probability space defined in Section 1. Define (θt) t∈ℝ on Ω by θtω=ω+tωt,tR,

then (Ω, ℱ, ℙ, (θt) t∈ℝ) is a metric dynamical system.

Set g~=(u~,v~,w~,y~)T, then the system (1)-(6) can be rewritten as follows: dg~dt=Ag~+F~g~+σg~dWtdt,t>0(10) g~0,x=g~x=u~x,v~x,w~x,y~xT,xD,(11)

where A=d1Δ0000d2Δ0000d1Δ0000d2Δ,F~g~=F+ku~+u~2v~+D1w~u~F1v~u~2v~+D2y~v~F+kw~+w~2y~+D1u~w~F1y~w~2y~+D2v~y~.

To transform the stochastic system into a deterministic system with random parameter, we introduce the following one-dimensional Ornstein-Uhlenbeck process: dz+zdt=dWt.(12)

From [13], we know that the stationary solution of Ornstein-Uhlenbeck process has the following form: zθtω0esθtωsds,tR.

Moreover, the random variable z(θt ω) is tempered, and ℙ—a.e. ω ∈ Ω, t ↦ z(θt ω) is continuous in t, and satisfies the properties (see [1, 11, 13]): limt±zθtωt=0;limt±1t0tzθsωds=0(13).

Set (ut,vt,wt,yt)T=eσzθtωu~t,v~t,w~t,y~tT Then, we obtain the equivalent system of (10) and (11) as: ut=d1ΔuF+kσzθtωu+e2σzθtωu2v+D1wu,(14) vt=d2Δv+Feσzθtω+σzθtωFve2σzθtωu2v+D2yv,(15) wt=d1ΔwF+kσzθtωw+e2σzθtωw2y+D1uw,(16) yt=d2Δy+Feσzθtω+σzθtωFye2σzθtωw2y+D2vy,(17)

that is, g = (u, v, w, y)T satisfies dgdt=Ag+Fg,ω+σzθtωg,t>0,(18) g0,x=g0x=eσzωg~0x=(u0x,v0x,w0x,y0x)T,xD,(19)

with Fg,w=F+ku+e2σzθtωu2v+D1wuFeσzθtωFve2σzθtωu2v+D2yvF+kw+e2σzθtωw2y+D1uwFeσzθtωFye2σzθtωw2y+D2vy.

Notice that for ℙ—a.e. ω ∈ Ω, F(g, ω) is locally Lipschtiz continuous with respect to g. In [10], You proved that the deterministic system has a unique solution by the Galekin method. Similar to deterministic system, by the Galekin method, for ℙ—a.e. ω ∈ Ω, we can prove that for g0 ∈ [L2(D)]4, (18)-(19) has a unique solution g(·, ω, g0) ∈ C([0, ∞), [L2(D)]4) ∩ L2((0, ∞), [H1(D)]4) with g(0, ω, g0) = g0. Moreover, similarly to Lemma 3 of [10], we can prove that g(t, ω, g0) is a unique, global, weak solution with respect to g0 ∈ [L2(D)]4, for t ∈ [0, ∞). This shows that (18) and (19) generate a continuous random dynamical system (φ(t))t ≥ 0 over (Ω, ℱ, ℙ, (θt) t∈ℝ) with φt,ω,g0=gt,ω,g0,t,ω,g0R+×Ω×[L2D]4.(20)

Now assume that ϕ : ℝ+ × Ω × [L2(D)]4 → [L2(D)]4 is given by ϕt,ω,g~0=g~t,ω,g~0=gt,ω,eσzωg0eσzθtω.(21)

Then ϕ is a continuous dynamical system associated to (10)-(11). Notice that two dynamical systems are conjugate to each other. Thus, in the following sections, we consider only the existence of a random attractor of φ.

3 Uniform estimates of solutions

To find the existence of the random attractor, we first need to obtain some uniform estimates of the solutions. Therefore in this section we first prove the uniform estimates about the solution of the two-compartment stochastic Gray-Scott equation on D, as t → + ∞. We assume that D is a collection of all tempered random subsets of [L2(D)]4. First, we define some functions which will be used in this section. Set Y1t,x=ut,x+vt,x+wt,x+yt,x,Y1,0=u0+v0+w0+y0;Y2t,x=ut,x+wt,x,Y2,0=u0+w0;Y3t,x=ut,x+vt,xwt,xyt,x,Y3,0=u0+v0w0y0;Y4t,x=vt,xyt,x;Y5t,x=ut,xwt,x.

The next lemma shows that φ has a random absorbing set in D.

Random dynamical system φ has a random absorbing set {K(ω)}ω∈Ω, in D, that is, for any {K(ω)}ω∈ΩD, and for ℙ — a.e. ω ∈ Ω, there is TB (ω) > 0 such that φ(t, θ—t ω, B(θ—t ω)) ⊂ K(ω), for any t > TB (ω).

 □

There exists a random variable ρ1 (ω), such that, for any B(ω) ∈ D, and g0 (ω) ∈ B(ω), for ℙ—a.e. ω∈ Ω, there is a TB (ω) > 0, such that, for any tTB (ω), the following estimate holds vt,θtω,g0θtωL66+yt,θtω,g0θtωL66ρ1ω.(52)

 □

There exists a random variable ρ2(ω) > 0 such that, for any B(ω)D,G0(ω)B(ω), for ℙ – a.e. ω ∈ Ω there exists a TB(ω) > 0 such that, for all tTB(ω), the following estimate holds, tt+1||g(s,θt1ω,g0(θt1ω))||2dsρ2(ω).(64)

 □

There exists a random variable ρ3(ω), such that, for any B(ω)D, and g0(ω), ∈ B(ω), for ω – a.e. ω ∈ Ω there is a TB(ω) > 0, such that for all tTB(ω), the following estimate holds, u(t,θtω,g0(θtω))2+w(t,θ1ω,g0(θ1ω)2ρ3(ω).

 □

There exists a random variable ρ4(ω), such that, for any B(ω)D and g0(ω), for ω – a.e. ω ∈ Ω there is a TB(ω) > 0, such that for all tTB(ω), the following estimate holds, υ(t,θtω,g0(θtω))2+y(t,θtω,g0(θtω))2ρ4(ω)(85)

 □

4 Existence of random attractors

In this section we use Proposition 2.1 to prove the existence of a pullback attractor.

The random dynamical system ϕ has a unique D–pullback random attractor in [L2(D)]4.

 □

Competing interests

The authors declare that there is no conflict of interest regarding the publication of this paper.

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About the article

Received: 2016-02-04

Accepted: 2016-06-25

Published Online: 2016-09-08

Published in Print: 2016-01-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0052.

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© 2016 Jia et al., published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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