Jiahe Xu,Kang Zhou,Qiuying Lu(Dept.of Math.,Zhejiang Sci-Tech University,Zhejiang 310018,PR China)

ASYMPTOTIC BEHAVIOR FOR GENERALIZED GINZBURG-LANDAU POPULATION EQUATION WITH STOCHASTIC PERTURBATION??

Jiahe Xu,Kang Zhou,Qiuying Lu?
(Dept.of Math.,Zhejiang Sci-Tech University,Zhejiang 310018,PR China)

In this paper,we are devoted to the asymptotic behavior for a nonlinear parabolic type equation of higher order with additive white noise.We focus on the Ginzburg-Landau population equation perturbed with additive noise. Firstly,we show that the stochastic Ginzburg-Landau equation with additive noise can be recast as a random dynamical system.And then,it is proved that under some growth conditions on the nonlinear term,this stochastic equation has a compact random attractor,which has a finite Hausdorff dimension.

Ginzburg-Landau model;additive white noise;random attractor;Hausdorff dimension

2000 Mathematics Subject Classification 35B40;35B41;37H10

1　Introduction

One of the most important problems in the fields of differential equations is that of the asymptotic behavior of evolution equations.During the last decades,finitedimension attractors for deterministic systems have been quite well investigated. Particularly in[8,9],the authors were devoted to the global attractors for noninvertible planar piecewise isometric maps and a class of nonhyperbolic piecewise affine maps.They obtained sufficient and necessary conditions for a compact set K to be the global attractor.Recently,Crauel and Flandoli[3]generalized the theory of deterministic attractors to the stochastic case.However,due to the introduction of random influences,the system are pushed out every bounded set with probability one.Therefore,we need to define the random attractor for the stochastic system. As far as we know,there are several different definitions,see[1,3,12,13].In[12,13],the authors considered the attractors for the Markov semigroup generated by a stochastic differential equation.While,in[1],they took the attractors as the ?-limit set for t→ ∞ of the trajectories.In this paper,we consider the attractors as a subset of the phase space (as in[3]) ,which is the ?-limit set at time t=0 of the trajectories“starting in bounded sets at time t=?∞.”For more detailed information about stochastic equations,one can refer to[5].

As we know,the famous Ginzburg-Landau model

was proposed in[4]for growth and dispersal in population.After that,there are a series of research on the existence,uniqueness and regularity of its global solutions, see[2,11].In this paper,we focus on the asymptotic behavior of the following nonlinear parabolic type equation of higher order perturbed by additive white noise

where a1＞0,a2＞0,D?Rnis a bounded open set with regular boundary?D, ν is the outward normal vector of the boundary?D,?j∈D (A) with j=1,···,m being time independent defined on D,andare independent two-sided realvalued Wiener processes on a complete probability space (?,F,P) ,

We assume that:

(H1)

(H2)

We present the following theorem of[3]for the existence of global attractor.

Theorem 1.1 Suppose that φ is an RDS on a Polish space X,and that there exists a compact ω→K (ω) absorbing every bounded nonrandom set B?X.Thenthe set

is a global attractor for φ,where ?B(ω) is the ?-limit set of B.Furthermore,A is measurable with respect to F if T is discrete,and it is measurable with respect to the completion of F (with respect to P) if T is continuous.

In order to further determine the bound for the random attractor,we apply the following theorem of[7].

Theorem 1.2 Let A (ω) ,ω∈?,be a compact measurable set invariant by a random map S (ω) ,ω∈?,such that

holds with an ergodic transformation θ.Assume that:

(i) S (ω) is almost surely uniformly differentiable on A (ω) ,it means that P?a.s, for every u∈A (ω) ,there exists a linear operator DS (ω,u) ∈L (H) ,the space of continuous linear operator from H to H,such that if u and u+h are in A (ω) ,then

where δ＞0,K (ω) is a random variable such that K (ω) ≥1 and E (lnK) ＜∞;

(ii) there exists an integrable random variablesatisfyingsuch thatfor any u∈A (ω) ,where

(iii) there exists a random variablesuch thatThen for almost all ω∈?,the Hausdorff dimension of A (ω) is less than d.

The whole paper is organized as follows.In Section 2,we give a brief introduction of Ornstein-Uhlenbeck process.In Section 3,we firstly define the stochastic flow ? (t,ω) associated with the stochastic Ginzburg-Landau equation (1) ,and then concentrate to get the existence of global attractor of the stochastic flow.In Section 4,we aim to establish the Hausdorff dimension of the random attractor.

2　Ornstein Uhlenbeck Process

Let α＞0 be given,for each j=1,2,···,m,zjbe the stationary (ergodic) solution of the one-dimensional equation

3　Existence of Random Attractor

Take the transformation v=u?z,then

In fact,by classical arguments as that of Theorem 1.1,Chap.III of[14],for P?a.s ω∈?,the following results hold:

(i) For all t0＜T∈R and all v0∈H,there exists a unique solution v∈

Thus,one can define a stochastic flow φ (t,ω) by

Note that by (H1) ,we have

while by Young inequality and (H2) ,there exist positive constants c1,c2,c3such that

Let

It follows that,for t＞s,

Lemma 3.1 There exists a random variable r?1(ω) such that for any B bounded in H,there exists an s0(B,ω) ,for any s＜s0(B,ω) ,we have

Since f (σ) has a subpolynomial growth (see[6],Lemma 15.4.4) ,for any B bounded in H,choosing an s (B,ω) →?∞such that for s＜s (B,ω) ,we have

The proof is complete.

Lemma 3.2 There exists a compact set K (ω) absorbing every bounded nonrandom set B∈H.

Proof Due to the former lemma,P?a.s,for any B bounded in H,there exists an s0(B,ω) such that for any s＜s0(B,ω) ,

Let u1and u2be two solutions of (1) .Subtracting the equations and then multiplying by u1?u2,we obtain

Thanks to (H2) ,there exists a c5such that

By Gronwall lemma,we obtain

Particularly,

Let {un(0) }n∈Nbe a sequence in K (ω) and vn(t) be a solution of equation (2) .From equation (3) ,for any t∈[?1,0],

By integrating on t∈[?1,0],one has

The proof is complete.

Now we are in position to present our result for the existence of random attractor by employing Theorem 1.1.

Theorem 3.1 The stochastic dynamical system φ (t,ω) generated by the nonlinear parabolic type equation of higher order with additive noise has a global attractor A (ω) ,which is measurable with respect to the completion of F (with respect to P) .

4　Hausdorff Dimension of A (ω)

Set S (ω) = φ (1,ω) ,then the random attractor A (ω) ,ω∈? is a compact measurable set invariant by S.

Lemma 4.1The mapping S (ω) is almost surely uniformly differentiable on A (ω) ,which means that for u,u+h∈A (ω) ,there exists a DS (ω,u) ∈L (H) ,such that

where δ＞0,K (ω) is a random variable satisfying K (ω) ≥1 and E (lnK) ＜∞.For any u0∈A (ω) ,DS (ω,u0) h=U (1) ,where U (t) is the solution of

and u (t) =φ (t,0,u0,ω) .

Proof Let u,u+h∈A (ω) ,and denote byfor t∈[0,1].let U be a solution of

and r=u1?u2?U,thus we get

Then following the similar calculations in[7],we have

Set

This insures that

where δ＞0,and K (ω) is a random variable satisfying K (ω) ≥1 and E (lnK) ＜∞.

The proof is complete.

Since α1(DS (ω,u) ) is equal to the norm of DS (ω,u) in L (H) ,we can choosewhich satisfies assumption (iii) of Theorem 1.2.

According to[14],

where U1,···,Udare d solutions of equation (5) satisfying h=ξi,and Qd(τ) = Qd(τ,u0;ξ1,···,ξd) is the orthogonal projector in H onto the space spanned by U1(τ) ,···,Ud(τ) .

At a given time τ,let φj(τ) ,j∈N,be an orthonormal basis of H,with φ1(τ) ,···,φd(τ) spanning Qd(τ) H=Span {U1(τ) ,···,Ud(τ) } ,then

we deduce that assumption (ii) of Theorem 1.2 holds provided that

In conclusion,we have the following Theorem 4.1.

Theorem 4.1 The stochastic nonlinear parabolic type equation of higher order (1) defines a stochastic semiflow φ (t,ω) which possesses a random attractor A (ω) . Moreover,the Haussdorff dimension of A (ω) is P?a.s finite and no larger than d defined by inequality (6) .

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(edited by Mengxin He)

?This work was under the grant of China Scholarship Council,National Natural Science Foundation of P.R.China (No.11101370,No.11302150,No.11211130093) ,the”521”talent program of Zhejiang Sci-Tech University (No.11430132521304) ,and Zhejiang Provincial Natural Science Foundation (LY13A010014) .

?Manuscript May 2,2015;Revised February 9,2016

?.E-mail:qiuyinglu@163.com