Three-Component Reversible Gray-Scott System on Infinite Lattices

Volume 2012, Article ID 340789,17pages doi:10.1155/2012/340789

Research Article

Random Attractors for Stochastic

Three-Component Reversible Gray-Scott System on Infinite Lattices

Anhui Gu,

Zhaojuan Wang,

and Shengfan Zhou

1College of Science, Guilin University of Technology, Guilin 541004, China

2School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China

3Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Correspondence should be addressed to Anhui Gu,[email protected] Received 13 April 2012; Accepted 29 May 2012

Academic Editor: Xiaohui Liu

Copyrightq2012 Anhui Gu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove the existence of a compact random attractor for the random dynamical system generated by stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices.

1. Introduction

Consider the following stochastic three-component reversible Gray-Scott system with a multiplicative white noise on infinite lattices:

du_i

dt d₁u_i−1−2u_iu_i1−Fkuiu²_iv_i−Gu³_i Nw_iu_i◦ dW dt , dv_i

dt d₂vi−1−2v_iv_i1 F1−v_i−u²_iv_iGu³_i v_i◦dW dt , dwi

dt d₃wi−1−2w_iw_i1 ku_i−FNwiw_i◦dW dt , u0 u0 ui0_i∈Z, v0 v0 vi0_i∈Z, w0 w0 wi0_i∈Z,

1.1

wherei∈Zthe set of integers,ui_i∈Z∈²,vi_i∈Z∈², andwi_i∈Z∈²; all the parameters are positive constants;W is a Brownian motion onΩ,F,Pand◦denotes the Stratonovich sense of the stochastic term.

System 1.1 can be considered as a discrete model of stochastic three-component reversible Gray-Scott system in which the existence of a random attractor has been established1. When there is no stochastic term, system1.1can be considered as a discrete analogue of the following three-component reversible Gray-Scott system in R:

∂u

∂t d1Δu−Fkuu²v−Gu³Nw,

∂v

∂t d₂ΔvF1−v−u²vGu³,

∂w

∂t d₃Δwku−FNw,

1.2

which was firstly introduced by Mahara et al.2, then it was reduced to system1.2under some nondimensional transformations in You3. Also, the existence of a global attractor for the solution semiflow of1.2with Neumann boundary condition on a bounded domain of space dimensionn≤3 was proved in3.

When G 0 and w 0, system 1.2 becomes the two-components Gray-Scott equations which was one of the models signified the seminal work of the Brussell school.

The model originated from describing an isothermal, cubic autocatalytic, continuously fed and diffusive reactions of two chemicals see 4–8, but neglected the reversible factors.

Indeed, the reversibility in the interactions of multispecies is an indispensable factor in many processes in natural and social sciences. If we take the reversibility into account, it yields system1.2.

Stochastic lattice differential equations have discrete spatial structures and take random influences into account. These random effects are not only introduced to compensate for the defects in some deterministic models, but are also rather intrinsic phenomena. Bates et al.9initiated the consideration of stochastic lattice dynamical systems with additive noises and Caraballo and Lu10was the first to consider the stochastic lattice dynamical systems with a multiplicative noise, and Han et al.11generalized the results of9,10to a more general space. For more details and the quite recent results, we can refer to, for example, 12–15.

Just like the models considered in biology, the discrete time models governed by difference equations are more appropriate than the continuous ones; we can also deal with the chemical and biochemical reactions in the same manner, see, for example,16,17and the references therein. However, very few investigations are on this topic, especially for the stochastic three-component reversible Gray-Scott system on infinite lattices, is widely open, to the best of our knowledge.

The paper is organized as follows. InSection 2, we present some preliminaries and definitions.Section 3is devoted to the existence of a random attractor.

2. Preliminaries

LetΩ,F,Pbe a probability space, whereΩis a subset ofC0R,R {ω∈ CR,R:ω0 0}, which endowed with the compact open topologysee 18,F is the Borel σ-algebra, andPis the corresponding Wiener measure on Ω. Letθtω· ω·t−ωt,t ∈ R, then Ω,F,P,θt_t∈Ris an ergodic metric dynamical system. Throughout the paper, we denote

²

x xi_i∈Z:xi∈R,

i∈Z

x²_i <∞

, 2.1

the Hilbert space equipped with the usual inner product and norm:

x, y

i∈Z

x_i, y_i

, x² x, x, ∀x x_ii∈Z, y y_i

i∈Z∈². 2.2

For the reader’s convenience, we introduce some basic concepts related to random dynamical systems and random attractor, which are taken from11,18,19. LetH, · Hbe a separable Hilbert space andΩ0,F,Pa probability space.

Definition 2.1. A stochastic process{ϕt, ω}_{t≥0,ω∈Ω0}is a continuous random dynamical system overΩ0,F,P,θt_t∈RifϕisB0,∞× F × BH,BH-measurable, and for allω∈Ω0,

ithe mappingϕt, ω:H→H,x→ϕt, ωxis continuous for everyt≥0, iiϕ0, ωis the identity onH,

iii cocycle propertyϕst, ω ϕt, θsωϕs, ωfor alls, t≥0.

Definition 2.2. iA set-valued mappingω→Bω⊂Hwe may write it asBωfor short is said to be a random set if the mappingω →dist_Hx, Bωis measurable for anyx∈H, where dist_Hx, Dis the distant inHbetween the elementxand the setD⊂H.

iiA random set Bω is said to be bounded if there existx0 ∈ H and a random variablerω>0 such thatBω⊂ {x∈H:x−x₀H≤rω, x0∈H}for allω∈Ω0.

iii A random set Bω is called a compact random set if Bω is compact for all ω∈Ω0.

ivA random bounded setBω⊂His called tempered with respect toθt_t∈Rif for a.e.ω∈Ω0, lim_t→∞e^−γtsup_{x∈Bθ−tω}xH 0 for allγ >0. A random variableω→rω∈R is said to be tempered with respect toθt_t∈Rif for a.e.ω ∈ Ω0, lim_t→∞sup_t∈Re^−γtrθ−tω 0 for allγ >0.

We consider a continuous random dynamical system RDS {ϕt, ω}_{t≥0,ω∈Ω0} over Ω0,F,P,θt_t∈RandDHthe set of all tempered random sets ofDH.

Definition 2.3. A random setKis called an absorbing set inDHif for allB∈ DHand a.e.

ω∈Ω0there existstBω>0 such that

ϕt, θ−tωBθ−tω⊂ Kω ∀t≥t_Bω. 2.3

Definition 2.4. A random set A is called a global random DH attractor pullback DH attractorfor{ϕt, ω}_{t≥0,ω∈Ω0}if the following hold:

iAis a random compact set, that is,ω→dx,Aωis measurable for everyx∈H andAωis compact for a.e.ω∈Ω0;

iiAis strictly invariant, that is, forω∈Ω0and allt≥0,ϕt, ωAω Aθtω;

iiiAattracts all sets inDH, that is, for allB∈ DHand a.e.ω∈Ω0, we have

tlim→∞d

ϕt, θ−tωBθ−tω,Aω

0, 2.4

wheredX, Y sup_x∈Xinf_y∈Yx−yHis the HausdorffsemimetricX⊆H, Y ⊆H.

Proposition 2.5see11. Suppose that

athere exists a random bounded absorbing setKω ∈ D²,ω ∈ Ω0, such that for any Bω ∈ D²and allω ∈ Ω0, there existsTω, B> 0 yieldingϕt, θ−tω, Bθ−tω ⊂ Kωfor allt≥Tω, B;

bthe RDS{ϕt, ω}_{t≥0,ω∈Ω0}is random asymptotically null onKω, that is, for any >0, there existT , ω, K>0 andI₀ , ω, K∈Nsuch that

sup

u∈Kω

|i|>I0 ,ω,Kω

ϕt, θ−tω, uθ−tω_i ²≤ ², ∀t≥T , ω, Kω. 2.5

Then the RDS{ϕt, ω,·}_{t≥0,ω∈Ω0}possesses a unique global randomDHattractor given by

Aω

τ≥Tω,K

t≥τ

ϕt, θ_−tω, Kθ_−tω. 2.6

3. Existence of a Random Attractor

In this section, we will derive the random attractor of the stochastic three-component reversible Gray-Scott lattice system1.1with a multiplicative white noise.

Forx xi_i∈Z, we defineA, B, B^∗ to be linear operators from² to² fori ∈ Z, as follows:

Ax_i−xi−12xi−x_i1, Bx_ix_i1−xi, B^∗x_ix_i−1−xi. 3.1

It is easy to show thatABB^∗B^∗B,B^∗x, x x, Bxfor allx, x∈², which implies that Ax, x≥0.

In the sequel, we rewrite the system 1.1 with initial values u0, v0, w0 : u0i, v_0i, w_0ii∈Z∈²×²×² Has the following integral equations inHfort≥0, ω∈Ω:

ut u_{0 t}

0

−d1Aus−Fkus u²svs

−Gu³s Nws ds

_t

0

us◦dW,

vt v0 _t

0

−d2Avs F1−vs−u²svs Gu³s ds

_t

0

vs◦dW,

wt w_{0 t}

0

−d3Aws kus−FNwsds _t

0

ws◦dW,

3.2

whereW is a two-sided Brownian motion on the same probability spaceΩ,F,P. To prove that this system 3.2 generates a random dynamical system, we will transform it into a random differential equation system inH.

Before performing this transformation, we need to recall some properties of the Ornstein-Uhlenbeck processes. Let

zθtω − ₀

−∞e^τθ_tωτdτ, t∈R, ω∈Ω. 3.3

We know thatzθtωis an Ornstein-Uhlenbeck process onΩ,F,P,θt_t∈Rand solves the following one-dimensional stochastic differential equationsee20for details:

z−zdtdwt, z−∞ 0, ∀t≥0, ω∈Ω, 3.4 wherewtω wt, ω ωtforω∈Ω, t∈R. In fact, we have the following.

Lemma 3.1see10,18. There exists aθ_t-variant setΩ ∈ FofC0R,Rof fullPmeasure such that, forω∈Ω, one has

ithe random variable|zω|is tempered;

iithe mapping

t, ω−→zθtω − ₀

−∞e^sωtsdsωt 3.5

is a stationary solution of Ornstein-Uhlenbeck equation3.4with continuous trajectories;

iii

t→ ±∞lim

|zθtω|

t 0, lim

t→ ±∞

1 t

_t

0

zθsωds0, 3.6

t→ ±∞lim 1 t

_t

0

|zθsω|dsE|z|<∞. 3.7

Obviously, e^zθtω1dH is clearly a homeomorphism in H, and the inverse operator e^−zθtω1dHis well defined. It easily follows from3.6thate^zθtω1dH_Hande^−zθtω1dH_H have subexponential growth ast → ±∞for allω∈Ω, which implies that they are tempered.

Since the mapping ofθonΩ has the same properties as the original one if we choose the trace σ-algebra with respect toΩ to be denoted also by F, we can change our metric dynamical system with respect toΩ, and still denoted by the symbols Ω,F,P,θt_t∈R.

Let

gt, ω e^−zθtω1dHgt, ω, ω∈Ω, 3.8 wheregt, ω ut, ω, vt, ω, wt, ω, and gt, ω ut, ω, vt, ω, wt, ωis a solution of 3.2. Then system 1.1 can be written as the following random system with random coefficients but without white noise:

du

dt −d1Au−Fk−zθtωue^2zθtωu²v−Ge^2zθtωu³Nw, dv

dt −d2AvFe^−zθtω−F−zθtωv−e^2zθtωu²vGe^2zθtωu³, dw

dt −d3Awku−FN−zθtωw

3.9

and an initial condition

g₀ω u0,v0, w0 u₀,v₀,w₀∈H. 3.10 Now we establish the following result.

Theorem 3.2. LetT >0 andg0ω∈Hbe fixed. Then the following properties hold:

ifor everyω∈Ω, system3.9admits a unique solutiong·, ω, g0∈ C0, T, H, iithe solutiongt, ω of system3.9depends continuously on the initial datag₀, that is, for

eachω∈Ω, the mappingg0∈H→g·, ω, g0∈ C0, T, His continuous.

Proof. 1Denote

L g

⎛

⎝−d1AFku 0 Nw

0 −d2AFv 0

ku 0 −d3AFNw

⎞

⎠,

Ψ g, θ_tω

⎛

⎝ zθtωue^2zθtωu²v−Ge^2zθtωu³ Fe^−zθtωzθtωv−e^2zθtωu²vGe^2zθtωu³

zθtωw

⎞

⎠.

3.11

Then system3.9can be written as dg

dt L g

Ψ g, θ_tω

. 3.12

Sincezθtωis continuous with respect tot, define

Ψ g, ω

⎛

⎝ zωue^2zωu²v−Ge^2zωu³ Fe^−zωzωv−e^2zωu²vGe^2zωu³

zωw

⎞

⎠, 3.13

then

Ψ g, ω

H ≤2F e^−zω 2

|zω|max{1, G}e^2zωg

H

31Ge^2zωg²

H.

3.14

For anyg u, v, w ui, v_i, w_ii∈Z,g u,v, w u_i,v_i,w_ii∈Z∈H, Ψ

g, ω

−Ψ g, ω

H≤2

zω|31Ge^2zωg²

Hg²

H

g−g

H. 3.15

For any bounded setD∈Hwith sup_g∈Dg_H≤r, define a random variableDωby

Dω 2

|zω|31Ge^2zωrmax{1, G}e^2zω

r2F e^−zω ≥0. 3.16

Then _τ1

τ

_Dθtωdt2 _τ1

τ

|zθtω|31Ge^2zθtωrmax{1, G}e^2zθtω

rFe^−zθtω dt

<∞, ∀τ ∈R,

3.17

and, for anyg, g,g∈D, we have Ψ

g, ω

H≤Dω, Ψ

g, ω

−Ψ g, ω

H≤Dωg−g

H. 3.18

Then we obtain thatLg Ψg, θ_tωis locally Lipschitz ing fromHtoH. By Proposition 2.1.1 in 19, problem3.12possesses a unique local solutiong·, ω,g0 ∈ C¹0, Tmax, H, where0, Tmaxis the maximal interval of existence of the solution of3.12. Now, we will show that the local solution is a global one. Define

rt N

k wt, μ k

N, 3.19

then system3.9can be written as du

dt −d1Au−Fk−zθtωue^2zθtωu²v−Ge^2zθtωu³kr, dv

dt −d2AvFe^−zθtω−F−zθtωv−e^2zθtωu²vGe^2zθtωu³, μdr

dt −μd3Arku−

μFk−zθtω r.

3.20

Taking the inner products of3.20withGut, vt andGwt, respectively, and adding up the resulting equalities, we get

1 2

d dt

Gu ²v ²μGr²

GFu²F

2v ²μGFr²

≤zθtω

Gu²v²Gr² F

2e^−2zθtω,

3.21

which implies that d dt

Gu²v² G μw ²

2GFu²Fv ²2GF μ w ²

≤2zθtω

Gu²v² G μ²w ²

Fe^−2zθtω.

3.22

Setting

λ Fmin

1,2G,2G/μ max

1, G, G/μ , C1 2 max

1, G, G/μ² min

1, G, G/μ , C2 F

min

1, G, G/μ, 3.23

then3.22yields d

dtgt²

H λ−C₁zθtωgt²

H≤C₂e^−2zθtω. 3.24

Applying Gronwall’s lemma to3.24, we obtain that, fort≥0, g

t, ω,g₀ω²

H≤e^{−λtC1t0zθsωds}g₀ω²

H

C2e^{−λtC10tzθsωds} _t

0

e^{−2zθsωλsC1s0zθτωdτ}ds.

3.25

Denoting

αω C_{1 T}

0

|zθsω|ds,

βω C2max

t∈0,T

e^{−λtC10tzθsωds} _t

0

e^{−2zθsωλsC1s0zθτωdτ}ds

,

3.26

we get

g

t, ω,g₀ω²

H≤g₀ω²

He^αωβω, 3.27

which implies that the solutiongis defined in any interval0, T.

2Letg₀,g₀∈Hand Xt:g

t, ω, g0

, Yt:g t, ω,g0

3.28 be the corresponding solutions of3.12. Then, denotingZt Z1, Z₂, Z₃ Xt−Yt u−u, v −v, w −w, we have

dZ₁

dt −d1AZ₁−Fk−zθtωZ1NZ₃e^2zθtω

u²v−u²v

−Ge^2zθtω

u³−u³ , dZ2

dt −d2AZ₂−F−zθtωZ2−e^2zθtω

u²v−u²v

Ge^2zθtω

u³−u³ , dZ3

dt −d3AZ3kZ1−FN−zθtωZ3.

3.29

Taking the inner product of3.30withZ1, Z2andZ3, respectively, it yields d

dt

Z1²Z2²Z3²

≤2zθtω

Z1²Z2²Z3² N

2 Z1²k 2Z3² 2e^2zθtω

u²v−u²v, Z ₁−Z₂

−2Ge^2zθtω

u³−u³, Z₁−Z₂ .

3.30

Due to3.28, we have e^2zθtω

u²v−u²v, Z 1−Z2

−Ge^2zθtω

u³−u³, Z1−Z2

≤21Ge^2zθtωg²

Hg²

H

Z1²Z2²

≤41Ge^2zθtωg²

H

Z1²Z2²Z3² .

3.31

Denotingδ 1/2max{N, k}, we obtain that d

dt

Z1²Z2²Z3²

≤

δzθtω 81Ge^2zθtωg²

H

Z1²Z2²Z3² .

3.32

By Gronwall’s lemma, fort∈0, T, we have Z1²Z2²Z3²≤

Z10²Z20²Z30

e^{δt81G0te2zθsωgs,ω,g0ω2Hds}, 3.33

whereδδmax_t∈0,T|zθtω|. According to3.26, fort∈0, T, _t

0

e^2zθsωg

s, ω,g0ω²

Hds

≤g0ω²

H

_t

0

e^{2zθsωC10szθτωdτ}ds C₂

_t

0

e^{2zθsωC10szθτωdτ} _s

0

e^{−2zθτω−C10tzθςωdς}dτds :g₀ω²

H

_t

0

ξθsωdsC_{2 t}

0

ξθsω _s

0

ξ⁻¹θτωdτds,

3.34

whereξθtω e^{2zθtωC10tzθτωdτ}. Obviously,t → lnξθtωis continuous P-a.s. Also, we have lim_t→∞lnξθtω/t 0, which implies that ξθtω is a tempered random variable.

Then by Proposition 4.3.3,18, for givenε >0, there is anε-slowly varying random variable Rωfor which

e^−ε|t|Rω≤ξθtω≤e^ε|t|Rω, ∀t∈R, ω∈Ω, 3.35

whereRω, ω∈Ωsatisfies

e^−ε|t|Rω≤Rθtω≤e^ε|t|Rω ∀t∈R, ω∈Ω. 3.36 Combining with3.35and3.36, we easily conclude that fort∈0, T,

_t

0

e^2zθsωg

s, ω,g₀ω²

Hds≤ g₀ω²

H

ε Rωe^εT C₂

2ε²e^2εT, 3.37

which implies sup

t∈0,TXt−Yt²≤

u0−u₀²v0−v₀²w0−w₀²

e^δT, 3.38

whereδδ81Gg0ω²_H/εTRωe^εT C2/2ε²Te^2εT<∞. Ifu0u0, v0v0and w₀w₀, then the above inequality shows the uniqueness and continuous dependence on the initial data of the solution of3.12. So the both results of the theorem hold.

Theorem 3.3. System3.12generates a continuous random dynamical systemϕt_t≥0 overΩ, F,P,θt_t∈R, whereϕt, ω, g0 gt, ω, g0forg₀ ∈H,t≥0 and for allω ∈Ω. Moreover, if one definesϕby

ψ t, ω, g0

e^zθtωϕ

t, ω, e^−zωg0

3.39

forg₀∈H,t≥0 and for allω∈Ω, thenψis another random dynamical system for which the process ω, t→ψt, ω, g0solves3.2for any initial conditiong0 ∈H.

Proof. The fact thatϕis continuous random dynamical system follows fromTheorem 3.2. The measurability ofψfollows from the properties of the transformationsee18,19. It follows directly the other statements.

Note that the two random dynamical systems are equivalent. It is easy to check that ψ has a random attractor provided ϕpossesses a random attractor. Then, we only need to consider the random dynamical systemϕ.

Now, we are in the position to study the existence of tempered random bounded absorbing set and global random attractor for the RDSϕinH.

Lemma 3.4. There exists a random bounded ballKω ∈ DHcentered at 0 with random radius ρω>0 such thatKω∈ DHis a random absorbing set forϕinDH; that is, for anyBω∈ DHandω∈Ω, there existsT_Bω>0 yieldingϕt, θ_−tω, Bθ_−tω⊂Kω, for allt≥T_Bω.

Proof. By substitutingωbyθ_−tωin3.26, we have g

t, θ_−tω,g₀θ_−tω²

H

≤e^{−λtC10tzθs−tωds}g0θ−tω²

HC2e^{−λtC10tzθs−tωds} _t

0

e^{−2zθs−tωλsC1s0zθτ−tωdτ}ds

≤e^{−λtC10tzθs−tωds}g₀θ_−tω²

HC_{2 0}

−te^{−2zθsωλs−tC1stzθτωdτ}ds

≤e^{−λtC1−t0zθsωds}g0θ−tω²

HC2

₀

−te^{−2zθsωλsC1s0zθτωdτ}ds

≤e^{−λtC1−t0zθsωds}g₀θ_−tω²

HC_{2 0}

−∞e^{−2zθsωλsC1s0zθτωdτ}ds.

3.40

By the properties of the Ornstein-Uhlenbeck process, ₀

−∞e^{−2zθsωλsC1s0zθτωdτ}ds <∞. 3.41 Notice that{Bω} ∈ DHis tempered, then forg0ω∈Bθ_−tω,

tlim→∞e^{−λtC1−t0zθsωds}g0θ−tω²

H0. 3.42

We can choose

ρ²ω 1C_{2 0}

−∞e^{−2zθsωλsC1s0zθτωdτ}ds, 3.43 thenKωis a random absorbing set forϕinDH, andKω∈ DH. Here, we remain only to check that

t→lim∞e^−γtρ²θ_−tω 0. 3.44

Indeed, obviously we have

e^−γtρ²θ−tω e^−γtC2e^−γt ₀

−∞e^{−2zθs−tωλsC1s0zθτ−tωdτ}ds !"e^−γt

→0 ast→∞

C₂e^−γt _−t

−∞e^{−2zθsωλsC1s−tzθτωdτ}ds ! "

→0 ast→∞

. 3.45

Lemma 3.5. The RDS{ϕt, ω,·}_{t≥0,ω∈Ω}generated by3.9is random asymptotically null onKω;

that is, for any >0, there existT , ω, K>0,∀t≥T , ω, Kω, andM , ω, Kω∈Nsuch that

sup

ϕ0∈Kω

|i|>M ,ω,Kω

ϕ

t, θ_−tω, ϕ0θ−tω

i ²

H ≤ ². 3.46

Proof. Choose a smooth cut-offfunction satisfying 0≤ ρs ≤1 fors ∈Randρs 0 for 0≤s≤1,ρs 1 fors≥2. Suppose there exists a constantcsuch that|ρs| ≤cfors∈R.

Setx ρ|i|/Mui_i∈Z, y ρ|i|/Mvi_i∈Z, andz ρ|i|/Mwi_i∈Z. By taking the inner product of3.20withGx,y, andGz, respectively, we get

G 2

d dt

i∈Z

ρ |i|

M

|ui|² −Gd1Au, x −GFk−zθtω

i∈Z

ρ |i|

M

|ui|² kGr, x Ge^2zθtω

u²v, x

−G²e^2zθtω u³, x

,

1 2

d dt

i∈Z

ρ |i|

M

|vi|² −d2

Av, y

−F−zθtω

i∈Z

ρ |i|

M

|vi|²

−Ge^2zθtω u²v, y

G²e^2zθtω u³, y

e^−zθtω, y , μG

2 d dt

i∈Z

ρ |i|

M

|r_i|² −μGd₃Ar, z−G

μFk−zθtω

i∈Z

ρ |i|

M

|r_i|² kGu, z.

3.47

Due to9,10, we have

−Au, x≤ 2cu²

M , −

Av, y

≤ 2cv²

M , −Ar, z≤ 2cr²

M . 3.48

Combining with3.48to3.49, we obtain 1

2 d dt

i∈Z

ρ |i|

M

G|ui|²|vi|²μG|ri|²

GF−zθtω

i∈Z

ρ |i|

M

|ui|²

F

2 −zθtω

i∈Z

ρ |i|

M

|vi|²G

μF−zθtω

i∈Z

ρ |i|

M

|ri|²

≤ 2c M

Gd1u ²d2v²μGd3r² F

2e^−2zθtω,

3.49

that is,

d dt

i∈Z

ρ |i|

M

G|u_i|²|v_i|²G μ|w_i|²

2GF

i∈Z

ρ |i|

M

|u_i|²

F

i∈Z

ρ |i|

M

|v_i|²2GF μ

i∈Z

ρ |i|

M

|w_i|²

≤2zθtω

i∈Z

ρ |i|

M

G|ui|²|vi|² G μ²|wi|²

4c M

Gd₁u²d₂v ²G μd₃w ²

Fe^−2zθtω.

3.50

Denote

C32cmax{d1, d2, d3}C1, 3.51

then3.51yields

d dt

i∈Z

ρ |i|

M gi ²

H λ−C1zθtω

i∈Z

ρ |i|

M gi ²

H≤ C₃ Mg²

HC2e^−2zθtω. 3.52

By using Gronwall’s lemma, fort≥T_KT_Kω, we have

i∈Z

ρ |i|

M gi

t, ω,g0ω ²

H

≤e^{−λt−TKC1}

t

Tkzθsωds

i∈Z

ρ |i|

M gi

Tk, ω,g0ω ²

H

C₃ M

_t

Tk

e^{−λt−sC1stzθτωdτ}g

s, ω,g0ω²

Hds C₂

_t

Tk

e^{−λt−s−2zθsωC1tszθτωdτ}ds.

3.53

Replaceωbyθ_−tω. We then estimate each term on the right-hand side of3.54. From3.26 withtreplaced byTKandωbyθ_−tω, respectively, it then follows that

e^{−λt−TKC1}

t

Tkzθs−tωds

i∈Z

ρ |i|

M g_i

T_k, θ_−tω,g₀θ_−tω ²

H

≤e^{−λtC10tzθs−tωds}g₀ω²

HC_{2 TK}

0

e^{−2zθs−tωλs−tC1tszθτ−tωdτ}ds

≤e^{−λtC1−t0zθsωds}g₀ω²

HC_{2 TK−t}

−t e^{−2zθsωλsC1s0zθτωdτ}ds.

3.54

Hence, by using3.6, there is aT₁ , ω, Kω> T_Kω, such that ift > T₁ , ω, Kω,

e^{−λt−TKC1}

t

Tkzθs−tωds

i∈Z

ρ |i|

M gi

Tk, θ_−tω,g0θ−tω ²

H <

2

3 . 3.55

Next, we estimate C₃

M _t

Tk

e^{−λt−sC1stzθτ−tωdτ}g

s, θ_−tω,g₀θ_−tω²

Hds

≤ C3

Mt−T_Ke^{−λtC10tzθr−tωdr}g₀ω²

HC2C3

M _t

TK

_s

0

e^{−2zθτ−tω−λt−τC1τtzθr−tωdr}dτds

≤ C₃

Mt−T_Ke^{−λtC1−t0zθrωdr}g₀ω²

HC₂C₃ M

_t

TK

_s−t

−t e^{−2zθτωλτC10τzθrωdr}dτds

≤ C₃

Mt−T_Ke^{−λtC1−t0zθrωdr}g₀ω²

HC₂C₃

M t−T_{K 0}

−te^{−2zθτωλτC1τ0zθrωdr}dτ.

3.56 By using3.7, there existT2 , ω, Kω > TKωandM1 , ω, Kω > 0, such that ift >

T₂ , ω, KωandM > M₁ , ω, Kω, then C3

M _t

Tk

e^{−λt−sC1stzθτ−tωdτ}g

s, θ_−tω,g0θ−tω²

Hds≤ ²

3. 3.57

By using3.7again, there existsT3 , ω, Kω>0, such that ifT > T3 , ω, Kω, we have

C_{2 t}

Tk

e^{−λt−s−2zθsωC1stzθτωdτ}ds≤ ²

3 . 3.58

Therefore, by letting

T , ω, Kω max{T1 , ω, Kω, T2 , ω, Kω, T3 , ω, Kω},

M , ω, Kω M1 , ω, Kω, 3.59

we obtain, fort > T , ω, KωandM > M , ω, Kω,

|i|≥2M

g_i

t, θ_−tω,g₀θ_−tω ²

H≤

i∈Z

ρ |i|

M g_i

t, θ_−tω,g₀θ_−tω ²

H≤ ², 3.60

which implies

|i|>M ,ω,Kω

ϕ

t, θ_−tω, ϕ₀θ−tω

i ²

H≤ ², ∀t≥T , ω, Kω. 3.61

The proof is completed.

Now, we have the main result.