# Existence of Random Attractors for ap-Laplacian-Type Equation with Additive Noise

Authors: Wenqiang Zhao; Yangrong Li
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

We first establish the existence and uniqueness of a solution for a stochasticp-Laplacian-type equation with additive white noise and show that the unique solution generates a stochastic
dynamical system. By using the Dirichlet forms of Laplacian and an approximation procedure, the nonlinear obstacle, arising from the additive noise is overcome when we make energy estimate. Then, we obtain a random attractor for this stochastic dynamical system. Finally, under a restrictive assumption on the monotonicity coefficient, we find that the random attractor consists of a single point, and therefore the system possesses a unique stationary solution.


---

## Body

## 1. Introduction

LetD⊂ℝn, n∈ℕ, be  a bounded open set with regular boundary ∂D. In this paper, we investigate the existence of a solution and a random attractor for the following quasilinear differential equation influenced by additive white noise(1.1)du+(Δ(Φp(Δu))+g(x,u))dt=f(x)dt+∑j=1mϕjdWj(t),x∈D,t≥0,
with the boundary conditions(1.2)∇u(t)=0,u(t)=0,x∈∂D,t≥0,
and the initial condition(1.3)u(0,x)=u0(x),x∈D.In (1.1), Φp(s)=|s|p-2s, p≥2, Wj(t)(1≤j≤m) are mutually independent two-sided real-valued Wiener processes, ϕj=ϕj(x)(1≤j≤m,x∈D) are given real-valued functions that will be assumed to satisfy some conditions. The unknown u(t) is a real-valued random process, sometimes denoted by u(t,x) or u(t,x,w). The exterior forced function g(x,s) defined in D×ℝ is subjected to the following growth and monotonicity assumptions:(1.4)g(x,s)s≥C1|s|q-Λ1(x),Λ1∈L1(D),C1∈R+,(1.5)|g(x,s)|≤C2|s|q-1+Λ2(x),Λ2∈Lq/(q-1)(D),C2∈R+,(1.6)(g(x,s1)-g(x,s2))(s1-s2)≥C3|s1-s2|2,C3∈R,
where 2≤q≤p<∞.In deterministic case (without random perturbed term), ifg(x,u)=ku, Temam [1] proved the existence and uniqueness of the solution, and then obtained the global attractor. Recently, Yang et al. [2, 3] obtained the global attractors for a general p-Laplacian-type equation on unbounded domain and bounded domain, respectively. Chen and Zhong [4] discussed the nonautonomous case where the uniform attractor was derived.It is  well known that the long-time behavior of random dynamical systems (RDS) is characterized by random attractors, which was first introduced by Crauel and Flandoli [5] as a generalization of the global attractors for deterministic dynamical system. The existence of random attractors for RDS has been extensively investigated by many authors, see [5–12] and references therein. However, most of these researches concentrate on the stochastic partial differential equations of semilinear type, such as reaction-diffusion equation [5–8], Ginzburg-Landau equation [9, 10], Navier-Stokes equation [5, 6], FitzHugh-Nagumo system [11] and so on. To our knowledge, recently, the Ladyzhenskaya model in [12] seems the first study on the random attractors for nonsemilinear type equations. It seems that the quasilinear type or complete nonlinear type evolution equations with additive noise take on severe difficulty when one wants to derive the random attractors.In this paper, we consider the existence and uniqueness of the solution and random attractor for (1.1) with forced term g(x,u) satisfying (1.4)–(1.6). The additive white noise ∑j=1mϕjdWj(t) characterizes all kinds of stochastic influence in nature or man-made complex system which we must take into consideration in the concrete situation.In order to deal with (1.1), we usually transform by employing a variable change the stochastic equation with a random term into a deterministic one containing a random parameter. Then the structure of the original equation (1.1) is changed by this transformation. As a result, some extra difficulties are developed in the process of the estimate of the solution, especially in the stronger norm space V, where V⊂H⊂V′ is the Gelfand triple; see Section 2. Hence, the methods (see [1–3]) used in unperturbed case are completely unavailable for obtaining the random attractors for (1.1).Though we also follow the classic approach (based on the compact embedding) widely used in [5, 6, 9, 10, 12] and so on, some techniques have to be developed to overcome the difficulty of estimate of the solution to (1.1) in the Sobolev space V. Fortunately, by introducing a new inner product over the resolvent of Laplacian, we surmount this obstacle and obtain the estimate of the solution in the Sobolev space V0, which is weaker than V, see Lemma 4.2 in Section 4. Here some basic results about Dirichlet forms of Laplacian are used. For details on the Dirichlet forms of a negative definite and self-adjoint operator please refer to [13]. The existence and uniqueness of solution, which ensure the existence of continuous RDS, are proved by employing the standard in [14]. If a restrictive assumption is imposed on the monotonicity coefficient in (1.6) we obtain a compact attractor consisting of a single point which attracts every deterministic bounded subset of H.The organization of this paper is as follows. In the next section, we present some notions and results on the theory of RDS and Dirichlet forms which are necessary to our discussion. In Section3, we prove the existence and uniqueness of the solution to the p-Laplacian-type equation with additive noise and obtain the corresponding RDS. In Section 4, we give some estimates for the solution satisfying (1.1)–(1.6) in given Hilbert space and then prove the existence of a random attractor for this RDS. In the last section, we prove the existence of the single point attractor under the given condition.


## 2. Preliminaries

In this section, we first recall some notions and results concerning the random attractor and the random flow, which can be found in [5, 6]. For more systematic theory of RDS we refer to [15]. We then list the Sobolev spaces, Laplacian and its semigroup and Dirichlet forms.The basic notion in RDS is a measurable dynamical system (MSD). The form(Ω,ℱ,P,θs) is called a MSD if (Ω,ℱ,P) is a complete probability space and {θs:Ω→Ω, s∈ℝ} is a family of measure-preserving transformations such that (s,w)↦θsw is measurable, θ0=id and θt+s=θtθs for all s,t∈ℝ.A continuous RDS on a complete separable metric space(X,d) with Borel sigma-algebra ℬ(X) over MSD (Ω,ℱ,P,θs) is by definition a measurable map (2.1)φ:R+×Ω×X⟶X,(t,w,x)⟼φ(t,w)x
such that P-a.s. w∈Ω(i)
φ(0,w)=id on X,(ii)
φ(t+s,w)=φ(t,θsw)φ(s,w) for all s,t∈ℝ+ (cocycle property),(iii)
φ(t,w):X→X is continuous for all t∈ℝ+.A continuous stochastic flow is by definition a family of measurable mappingS(t,s;w):X→X,-∞≤s≤t≤∞, such that P-a.s. w∈Ω(2.2)S(t,r;w)S(r,s;w)x=S(t,s;w)x,x∈X,S(t,s;w)x=S(t-s,0.;θsw)x,x∈X,
for all s≤r≤t, and s↦S(t,s;w)x is continuous in X for all s≤t and x∈X.A random compact set{K(w)}w∈Ω is a family of compact sets indexed by w such that for every x∈X the mapping w↦d(x,K(w)) is measurable with respect to ℱ.Let𝒜(w) be a random set. One says that 𝒜(w) is attracting if for P-a.s. w∈Ω and every deterministic bounded subset B⊂X(2.3)lim⁡t→∞dist⁡(φ(t,θ-tw)B,A(w))=0,
where dist(·,·) is defined by dist(A,B)=supx∈Ainfy∈Bd(x,y).We say that𝒜(w) absorbs B⊂X if P-a.s. w∈Ω, there exists tB(w)>0 such that for all t≥tB(w), (2.4)φ(t,θ-tw)B⊂A(w).Definition 2.1.
Recall that a random compact setw↦𝒜(w) is called to be a random attractor for the RDS φ if for P-a.s. w∈Ω(i)
𝒜(w) is invariant, that is, φ(t,w)𝒜(w)=𝒜(θtw), for all t≥0;(ii)
𝒜(w) is attracting.Theorem 2.2 (see [5]).
Letφ(t,w) be a continuous RDS over a MDS (Ω,ℱ,P;θt) with a separable Banach Space X. If there exists a compact random absorbing set K(w) absorbing every deterministic bounded subset of X, then φ possesses a random attractor 𝒜(w) defined by
(2.5)A(w)=⋃B∈B(X)⋂s≥0⋃t≥sφ(t,θ-tw)B¯¯,
where ℬ(X) denotes all the bounded subsets of X.LetLp(D) be the p-times integrable functions space on D with norm denoted by ∥·∥p, 𝒱(D) be the space consisting of infinitely continuously differential real-valued-functions with a compact support in D. We use V to denote the norm closure of 𝒱(D) in Sobolev space W2,p(D), that is, V=W02,p(D). Since D is a bounded smooth domain in ℝn, we can endow the Sobolev space V with equivalent norm (see [1, page 166]) (2.6)‖v‖V=‖Δv‖p=(∫D|Δv|pdx)1/p,v∈V.
Define V′ = the dual of V, that is, V′=W-2,p′(D). Then we have (2.7)T∈W-2,p′(D)⟺T=∑|α|≤2Dαfα,fα∈Lp′(D),
where 1/p+1/p′=1. Let H denote the closure of L2(D) in 𝒱(D) with the usual scalar product and norm {(·,·),∥·∥2}. Identifying H with its dual space H′ by the Riesz isomorphism i:H→H′, we have the following Gelfand triple: (2.8)V⊂H≡H′⊂V′,
or concretely (2.9)W02,p(D)⊂L2(D)⊂(W02,p(D))′=W-2,p/(p-1)(D),
where the injections are continuous and each space is dense in the following one.We define the linear operatorA by Au=Δu for u∈H01(D)⋂H2(D). Then A is negative definite and self-adjoint. It is well-known that A (with domain W02,p(D)) generates a strongly continuous semigroup M(t) on Lp(D) which is contractive and positive. Here “contractive” means ∥M(t)∥p≤1 and “positive” means M(t)u≥0 for every 0≤u∈Lp(D). The resolvent of generator A denoted by R(λ,A), λ∈ρ(A), where ρ(A) is the resolvent set of A. By Lumer-Phillips Theorem in [16], it follows that (0,∞)⊂ρ(A) and for u∈Lp(D)(2.10)R(λ,A)u=(λ-A)-1u=∫0∞e-λtM(t)udt,λ>0,(2.11)M(t)u=lim⁡n→∞[ntR(n/t,A)]nu,t>0.
Furthermore, by (2.10) for every u∈Lp(D) we have (2.12)‖λR(λ,A)u‖p≤‖u‖p,λ>0,λR(λ,A)u⟶u,asλ→∞,
where the convergence is in the Lp-norms. Moreover, for u∈D(A), it follows that R(λ,A)u∈D(A), and AR(λ,A)u=R(λ,A)Au.SinceA is negative definite and self-adjoint operator on H01(D)⋂H2(D), we associate A with the Dirichlet forms  [13] ɛ by (2.13)ɛ(u,v)=(-Au,-Av),u,v∈H01(D).ɛ is unique determined by A. For u,v∈H01(D), we define a new inner product by(2.14)ɛ(λ)(u,v)=λ(u-λR(λ,A)u,v),λ>0,
where R(λ,A) is the resolvent of A. Then, we have the basic fact (see [13]) that ɛ(λ)(u,v)↑ as λ→∞, and(2.15)lim⁡λ→∞ɛ(λ)(u,v)=ɛ(u,v),
for u,v∈H01(D).


## 3. Existence and Uniqueness of RDS

We introduce an auxiliary Wiener process, which enables us to change (1.1) to a deterministic evolution equation depending on a random parameter. Here, we assume that W(t) is a two-sided Wiener process on a complete probability space (Ω,ℱ,P), where Ω={w∈C(ℝ,ℝm):w(0)=0}, ℱ is the Borel sigma-algebra induced by the compact-open topology of Ω and P is the corresponding Wiener measure on (Ω,ℱ). Then we have (3.1)w(t)=W(t)=(W1(t),W2(t),…,Wm(t)),t∈R.
Define the time shift by (3.2)θtw(s)=w(s+t)-w(t),w∈Ω,t,s∈R.
Then (Ω,ℱ,P,θt) is a ergodic measurable dynamical system.In order to obtain the random attractor, in our following discussion, we always assume thatϕj(1≤j≤m) belong to W04,p(D) and ∇ϕj(1≤j≤m)=0.We now employ the approach similar to [5] to translate (1.1) by one classical change of variables(3.3)v(t)=u(t)-z(t),
where, for short, z(t)=z(t,w)=∑j=1mϕjWj(t). Then, formally, v(t) satisfies the following equation parameterized by w∈Ω:(3.4)dvdt+Δ(Φ(Δv+Δz))+g(x,v+z)=f(x),x∈D,t≥s,(3.5)v(s)=u(s)-z(s),x∈D,s∈R,(3.6)∇v=0,v=0,x∈∂D,t≥s,
where g(x,u) satisfies (1.4)–(1.6) and f is given in V′, 2≤q≤p<∞.We define a nonlinear operatorΨ on V(3.7)Ψ(v)=Δ(Φp(Δ(v+z)))+g(x,v+z)-f(x),
for v∈V, x∈D. Then we have(3.8)Ψ(v)=Ψ¯(u),
where we define Ψ¯(u)=Δ(Φp(Δu))+g(x,u)-f(x) with u=v+z as in (3.3). So we can deduce problem (1.1) to the problem(3.9)dvdt+Ψ(v)=0,t≥s,
with initial condition v(s)=u(s)-z(s) for s∈ℝ. Moreover, by (3.9), it follows that the solutions w-wise satisfy the following: (3.10)v(t)=v(s)-∫stΨ(v(τ))dτ,
with v(s)=u(s)-z(s) and t≥s.Sincep≥q, by our assumption (1.4)–(1.6) and f∈V′, it is easy to check that the operator Ψ:v↦Ψ(v) mapping W2,p(D) into W-2,p′(D) is well-defined, where p′=p/(p-1). We now prove the existence and uniqueness of solution to (3.4).Theorem 3.1.
Assume thatg satisfies (1.4)–(1.6) and f is given in V′, 2≤q≤p<∞. Then for all s∈ℝ and v0∈H with v0=v(s), (3.4) has a unique solution
(3.11)v(t,w;s,v0)∈Lloc⁡p([s,∞),V)⋂C([s,∞),H)
for t≥s and P-a.s. w∈Ω. Furthermore, the mapping v0↦v(t,w;s,v0) from H into H is continuous for all t≥s.Proof.
We will show thatΨ(v) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. Then for every v0∈H with v0=v(s), the existence and uniqueness of solution v(t)=v(t,w;s,v0)∈Llocp([s,∞),V) follow from [13, Theorem 4.2.4]. If the solution v∈Lp([s,T],V), T>0, then it is elementary to check that Ψ(v) belongs to Lp′([s,T],V′) by our assumption p≥q and f∈V′. Thus, from (3.9), we get that vt∈Lp′([s,T],V′). Now by the general fact (see [1, page 164, line 1–3]) it follows that v is almost everywhere equal to a function belonging to C([s,T],H). The continuity of the mapping v0↦v(t,w;s,v0) from H into H is easily proved by using the monotonicity of Ψ.
By [13, Theorem 4.2.4], it remains to show that Ψ(v) possesses hemicontinuity, monotonicity, coercivity, and boundedness properties. For convenience of our discussion in the following, we decompose Ψ(v)=Ψ1(v)+Ψ2(v), where Ψ1(v)=Δ(Φp(Δu)) and Ψ2(v)=Ψ(v)-Ψ1(v), where Ψ is as in (3.7).
We first prove the hemicontinuity, that is, for everyv1,v2,v3∈V, the function λ→(Ψ(v1+λv2),v3) is continuous from ℝ→ℝ. But it suffices to prove the continuity at λ=0. So we assume that |λ|<1. For v1,v2,v3∈V, by integration by parts, we see that(3.12)(Ψ1(v1+λv2),v3)=∫D|Δ(v1+λv2+z)|p-2Δ(v1+λv2+z)Δv3dx.
By Hölder's inequality and Young's inequality, it yields that
(3.13)||Δ(v1+λv2+z)|p-2Δ(v1+λv2+z)Δv3|≤|Δ(v1+λv2+z)|p-1|Δv3|≤2p-2(|Δ(v1+z)|p-1+|Δv2|p-1)|Δv3|≤2p-2(|Δ(v1+z)|p+|Δv2|p+2|Δv3|p),
which the right-hand side is in L1(D) for v1,v2,v3∈V. Hence the expression of the right-hand side of inequality (3.13) is the control function for the integrant in (3.12). Then the Lebesgue's dominated convergence theorem can be apply to (3.12) when we take the limit λ→0. This proves the hemicontinuity of Ψ1(v). As for the hemicontinuity of Ψ2(v), noting that by our assumption (1.5) and f∈V′ we have
(3.14)(Ψ2(v1+λv2),v3)=∫Dg(x,v1+λv2+z)v3dx-∫Df(x)v3dx≤C2∫D|v1+λv2+z|q-1|v3|dx+∫DΛ2(x)|v3|dx-∫Df(x)v3dx.
It suffices to find the control function for the first integrand above, but we can get this by noting that q≤p and using approach similar to (3.13).
Second, we prove the monotonicity ofΨ(v). We first prove the monotonicity for Ψ1. For v1,v2∈V, since v1=u1-z, v2=u2-z, we have(3.15)(Ψ1(v1)-Ψ1(v2),v1-v2)=(|Δu1|p-2Δu1-|Δu2|p-2Δu2,Δu1-Δu2)=∫D(|Δu1|p+|Δu2|p-|Δu1|p-2Δu1Δu2-|Δu2|p-2Δu2Δu1)dx≥∫D(|Δu1|p+|Δu2|p-|Δu1|p-1|Δu2|-|Δu2|p-1|Δu1|)dx=∫D(|Δu1|p-1-|Δu2|p-1)(|Δu1|-|Δu2|)dx≥0.
Since p≥2, the function sp-1 is increasing for s≥0, which shows that the last inequality in the above proof is correct. On the other hand, by our assumption (1.6), we have (Ψ2(v1)-Ψ2(v2),v1-v2)≥C3∥v1-v2∥22, and therefore it follows that for v1,v2∈V(3.16)(Ψ(v1)-Ψ(v2),v1-v2)≥C3‖v1-v2‖2,
where C3 is as in (1.6). Hence, we have showed the monotonicity of Ψ(v).
As for the coercivity, forv∈V, by our assumptions (1.4) and (1.5), using Hölder' inequality, we have(3.17)(Ψ(v),v)=∫DΔ(|Δu|p-2Δu)vdx+∫Dg(x,v+z)vdx-∫Df(x)vdx=‖Δu‖pp-∫D(|Δu|p-2Δu)Δzdx+∫Dg(x,u)udx-∫Dg(x,u)zdx-∫Df(x)vdx≥‖Δu‖pp-‖Δu‖pp-1‖Δz‖p+C1‖u‖qq-‖Λ1‖1-C2‖u‖qq-1‖z‖q-‖Λ2‖q′‖z‖q-‖f‖V′‖v‖V,
where C1 and C2 are defined in (1.4) and (1.5). By employing the ɛ-Young's inequality, that is, ab≤ɛ(ar/r)+ɛ-r′/r(br′/r′) for r>1 and 1/r+1/r′=1, we find that
(3.18)‖Δu‖pp-1‖Δz‖p≤14p-1p‖Δu‖pp+22p-2p‖Δz‖pp≤14‖Δu‖pp+22p-2‖Δz‖pp.
Similarly, we have
(3.19)C2‖u‖qq-1‖z‖q≤C12‖u‖qq+2q-1C11-qC2q‖z‖qq,(3.20)‖f‖V′‖v‖V≤14‖Δu‖pp+22/(p-1)‖f‖V′p/(p-1)+‖f‖V′‖Δz‖p,
then, by (3.17)–(3.20), we obtain that
(3.21)(Ψ(v),v)≥12‖Δu‖pp+C12‖u‖qq-p1(t,w),
with
(3.22)p1(t,w)=22p-2‖Δz‖pp+2q-1C11-qC2q‖z‖qq+‖Λ2‖q′‖z‖q+22/(p-1)‖f‖V′p/(p-1)+‖f‖V′‖Δz‖p≥0,
where q′ is the dual number of q. At the same time, (3.21) is one form of coercivity which will be used in Section 4, but in order to prove the existence and uniqueness of solution to (3.4), we will give another form.
Noting that by Hölder's inequality it follows withu(t)=v(t)+z(t) that(3.23)‖u‖qq=‖v(t)+z(t)‖qq≥21-q‖v‖qq-‖z‖qq,
then by the inverse ɛ-Young's inequality, that is, ab≥ɛ(ar/r)+ɛ-r′/r(br′/r′) when r<1 and 1/r+1/r′=1, we get from (3.23) that
(3.24)‖u‖qq≥21-q‖v‖qq-‖z‖qq≥21-qη0q‖v‖2q-‖z‖qq≥q2‖v‖22-‖z‖qq-C′,
where C′=((q-2)/2)2(2-2q)/(2-q)η02q/(2-q) and η0 is the Sobolev embedding coefficient of Lq(D)↪L2(D). Hence, it follows from (3.21) that
(3.25)(Ψ(v),v)≥12‖Δu‖pp+q2‖v‖22-p2(t,w),
with p2(t,w)=p1(t,w)+∥z∥qq+C′≥0, where p1(t,w) is defined as in (3.22). Note that if q=2, we omit this procedure and directly (3.21) passes to (3.25). Hence we have proved the coercivity for Ψ.
We finally prove the the boundedness forΨ(v) for fixed v∈V, that is, for fixed v∈V, Ψ(v) is a linear bounded functional on W02,p(D). Indeed, for v,h∈V, by applying Hölder's inequality and repeatedly using Sobolev's embedding inequality, we have(3.26)(Ψ(v),h)≤∫D|Δu|p-1|Δh|dx+∫D|g(x,u)||h|dx+∫D|f(x)||h|dx≤‖Δu‖pp-1‖Δh‖p+C2‖u‖qq-1‖h‖q+‖Λ2‖q′‖h‖q+‖f‖V′‖h‖V≤(‖Δu‖pp-1+c1‖Δu‖pq-1+c2‖Λ2‖q′+‖f‖V′)‖Δh‖p≤(2‖Δu‖pp-1+c3+c2‖Λ2‖q′+‖f‖V′)‖Δh‖p≤(2p-1‖Δv‖pp-1+p3(t,w))‖Δh‖p
with  the random variable p3(t,w)=2p-1∥Δz∥pp-1+c2∥Λ2∥q′+c2∥f∥V′+c3≥0 and the positive constants ci(i=1,2,3) independent of v,h. Therefore, from (3.26) we finally find that
(3.27)‖Ψ(v)‖V′≤2p-1‖Δv‖pp-1+p3(t,w),
is a bounded linear operator on W02,p(D) for fixed v∈V. From the proof we know that the assumption p≥q≥2 is necessary. This completes the proof of Theorem 3.1.We now define(3.28)S(t,s;w)u0=v(t,w;s,u0-z(s,w))+z(t,w),t≥s∈R,
with u0=u(s). Then S(t,s;w)u0 is the solution to (1.1) in certain meaning for every u0∈H and t≥s∈ℝ. By the uniqueness part of solution in Theorem 3.1, we immediately get that S(t,s,w) is a stochastic flow, that is, for every u0∈H and t≥r≥s∈ℝ(3.29)S(t,s;w)u0=S(t,r;w)S(r,s;w)u0,(3.30)S(t,s;w)u0=S(t-s,0;θsw)u0.
Hence if we define(3.31)φ(t,w)u0=S(t,0;w)u0=v(t,w;0,u0-z(0,w))+z(t,w)
with u0=u(0), then by Theorem 3.1  φ is a continuous stochastic dynamical system associated with quasilinear partial differential equation (1.1), with the following fact(3.32)φ(t,θ-tw)u0=u(0,w;-t,u0),∀t≥0,
that is to say, φ(t,θ-tw)u0 can be interpreted as the position of the trajectory at time 0, which was in u0 at time -t (see [5]).


## 4. Existence of Compact Random Attractor for RDS

In this section, we will compute some estimates in spaceH=L2(D) and V0=H01(D). Note that pi(t,w)(i=1,2,3) appearing in the proofs are given in Section 3. In the following computation, w∈Ω; the results will hold for P-a.s. w∈Ω.Lemma 4.1.
Suppose thatg satisfies (1.4)–(1.6) and f is given in V′. Then there exist random radii r1(w),r2(w)>0, such that for all ρ>0 there exists s=s(w,ρ)≤-1 such that for all s≤s(w,ρ) and all u0∈H with ∥u0∥2≤ρ, the following inequalities hold for P-a.s. w∈Ω(4.1)‖v(t,w;s,u0-z(s))‖22≤r12(w),∀t∈[-1,0],∫-10(‖Δu(τ,w;s,u0)‖pp+‖u(τ,w;s,u0)‖qq)dτ≤r22(w),
where v(t,w;s,u0-z(s)) is the solution to (3.4) with v(t,w;s,u0-z(s))=u(t,w;s,u0)-z(t,w) and u0=u(s).Proof.
For simplicity, we abbreviatev(t):=v(t,w;s,u0-z(s)) and u(t):=u(t,w;s,u0) for fixed u0,w∈Ω and t≥s with u0=u(s). Multiplying both sides of (3.9) by v(t) and then integrating over D, we obtain that
(4.2)12ddt‖v‖22+(Ψ(v),v)=0.
Then, by (3.25), we have
(4.3)ddt‖v‖22+‖Δu‖pp+q‖v‖22≤2p2(t,w).
Applying the Gronwall's lemma to (4.3) from s to t, t∈[-1,0], it yields that
(4.4)‖v(t)‖22≤e-q(t-s)‖v(s)‖22+2∫stp2(τ,w)e-q(t-τ)dτ≤2eq(eqs‖u0‖22+eqs‖z(s)‖22+∫-∞0p2(τ,w)eqτdτ),
where p2(τ,w) grows at most polynomially as τ→-∞ (see [5]). Since p2(τ,w) is multiplied by a function which decays exponentially, the integral in (4.4) is convergent.
Given every fixedρ>0, we can choose s(w,ρ), depending only on w and ρ, such that eqs∥u0∥22≤1. Similarly, ∥z(s)∥22 grows at most polynomially as s→-∞, and ∥z(s)∥22 is multiplied by a function which decays exponentially. Then we have (4.5)sup⁡s≤0eqs‖z(s)‖22<+∞.
Hence by (4.4) we can give the final estimate for ∥v(t)∥22(4.6)‖v(t)‖22≤r12(w):=2eq(1+sup⁡s≤0eqs‖z(s)‖22+∫-∞0p2(τ,w)eqτdτ),
for t∈[-1,0]. Following (4.2), by using (3.21), we find that
(4.7)ddt‖v‖22+‖Δu‖pp+C1‖u‖qq≤2p1(t,w),
where p1(t,w) is the same as in (3.22). Integrating (4.7) for t on [-1,0], we get that
(4.8)∫-10‖Δu(τ)‖pp+C1‖u(τ)‖qqdτ≤2∫-10p1(τ,w)dτ+‖v(-1)‖22,
which gives an expression for r22(w).In the following, we give the estimate of∥∇u(t)∥2. This is the most difficult part in our discussion. Because the nonlinearity of Ψ1 and Ψ2 in (3.4) or (3.9), it seems impossible to derive the V-norm estimate by the way as [1, page 169]. So we relax to bound the solution in a weaker Sobolev V0=H01(D) with equivalent norms denoted by ∥∇u∥2 for u∈V0. Here, just as our statement in the introduction, we make the inner product over the resolvent R(λ,A) which is defined in Section 2, then by using the Dirichlet forms of A we obtain technically the estimate of ∥∇u(t)∥2.Lemma 4.2.
Suppose thatg satisfies (1.4)–(1.6) and f is given in V′. Then there exists a random radius r3(w)>0, such that for all ρ>0 there exists s=s(w,ρ)≤-1 such that for all s≤s(w,ρ) and all u0∈H with ∥u0∥2≤ρ, the following inequality holds for P-a.s. w∈Ω(4.9)‖∇u(t,w;s,u0)‖22≤r32(w),∀t∈[-1,0],
where u(t,w;s,u0) is the solution to (1.1) with u0=u(s). In particular,
(4.10)‖∇u(0,w;s,u0)‖22≤r32(w).Proof.
Taking the inner product of (3.9) with -λAR(λ,A)v where λ>0, v∈V, we get
(4.11)-∫DvtλAR(λ,A)vdx=∫DΨ1(u)λAR(λ,A)vdx+∫DΨ2(u)λAR(λ,A)vdx.
By the semigroup theory (see [16]) we have
(4.12)AR(λ,A)v=R(λ,A)Av=λR(λ,A)v-v,
for v∈D(A). We now estimate every terms on the right-hand side of (4.11). The first term on the right-hand side of (4.11) is rewritten as
(4.13)∫DΨ1(u)λAR(λ,A)vdx=∫DΨ1(u)λAR(λ,A)udx-∫DΨ1(u)λAR(λ,A)zdx.
Employing (4.12) and by integration by parts, it yields that
(4.14)∫DΨ1(u)λAR(λ,A)udx=λ∫DΨ1(u)(λR(λ,A)u-u)dx=-λ∫DΔ(|Δu|p-2Δu)udx+λ∫DΔ(|Δu|p-2Δu)λR(λ,A)udx=-λ‖Δu‖pp+λ∫D(|Δu|p-2Δu)λΔR(λ,A)udx≤-λ‖Δu‖pp+λ∫D|Δu|p-1|λR(λ,A)Δu|dx≤-λ‖Δu‖pp+λ‖Δu‖pp-1‖λR(λ,A)Δu‖p≤-λ‖Δu‖pp+λ‖Δu‖pp=0,
where we use the contraction property of λR(λ,A) on Lp(D), that is, ∥λR(λ,A)Δu∥p≤∥Δu∥p for Δu∈Lp(D) and every λ>0. We now bound the second term on the right-hand side of (4.13)
(4.15)-∫DΨ1(u)λAR(λ,A)zdx≤‖Ψ1(u)‖V′‖λAR(λ,A)z‖V=‖Ψ1(u)‖V′‖λR(λ,A)Az‖V,
where we use our assumption ϕj(1≤j≤m)∈W04,p(D). Since Ψ1 maps V into V′, then for fixed u∈V and every h∈V, we have
(4.16)(Ψ1(u),h)=∫DΔ(|Δu|p-2Δu)hdx=∫D(|Δu|p-2Δu)Δhdx≤∫D|Δu|p-1|Δh|dx≤‖Δu‖pp-1‖Δh‖p.
So for fixed u∈V, ∥Ψ1(u)∥V′≤∥Δu∥pp-1, and therefore by (4.15) we obtain that
(4.17)-∫DΨ1(u)λAR(λ,A)zdx≤‖Δu‖pp-1‖λR(λ,A)Az‖V≤C‖Δu‖pp-1‖Az‖V≤‖Δu‖pp+Cp‖Az‖Vp,
where ∥λR(λ,A)Az∥V≤C∥Az∥V and C is a constant independent of λ, v(t) and u(t). Here we should note that λR(λ,A) is a bounded linear operator on V. Hence, by (4.14)–(4.17) the first term on the right-hand side of (4.11) is finally bounded by
(4.18)∫DΨ1(u)λAR(λ,A)vdx≤‖Δu‖pp+Cp‖Az‖Vp.
By our assumption (1.5), the second term on the right-hand side of (4.11) is estimated as
(4.19)∫DΨ2(u)λAR(λ,A)vdx=∫D(g(x,u)-f(x))λR(λ,A)Avdx≤∫D|g(x,u)||λR(λ,A)Av|dx+∫D|f(x)||λAR(λ,A)v|dx≤∫D(C2|u|q-1+Λ2(x))|λR(λ,A)Av|dx+∫D|f(x)||λR(λ,A)Av|dx≤C2‖u‖qq-1‖λR(λ,A)Av‖q+‖Λ2‖q′‖λR(λ,A)Av‖q+‖f‖2‖λR(λ,A)Av‖2≤C2‖u‖qq-1‖Av‖q+‖Λ2‖q′‖Av‖q+‖f‖2‖Av‖2≤‖u‖qq+(C2q+1)‖Av‖qq+‖Λ2‖q′q′+‖f‖22+‖Av‖22,
where we employ Young's inequality ab≤ar+br/(r-1) for r>1. But, by Sobolev's inequality and Young's inequality, it yields that
(4.20)‖Av‖qq≤η1q‖Av‖pq≤η1p‖Av‖pp+1≤2p-1η1p‖Au‖pp+2p-1η1p‖Az‖pp+1;
similarly
(4.21)‖Av‖22≤2p-1η2p‖Au‖pp+2p-1η2p‖Az‖pp+1,
where the positive constants η1,η2 are Sobolev's embedding constants independent of λ. Then by (4.19)–(4.21), there exist positive constants c1,c2 such that
(4.22)∫DΨ2(u)λAR(λ,A)vdx≤‖u‖qq+c1‖Au‖pp+c2‖Az‖pp+‖Λ2‖q′q′+‖f‖22+2,
where q′=q/(q-1). By (4.18) and (4.22), we find that (4.11) becomes
(4.23)-∫DvtλAR(λ,A)vdx≤c3‖Δu‖pp+‖u‖qq+p4(t,w),
where p4(t,w)=Cp∥Az∥Vp+c2∥Az∥pp+∥Λ2∥q′q′+∥f∥22+2≥0 and c3=c1+1. On the other hand, by (4.12) and the Dirichlet forms (2.14), we have
(4.24)-∫DvtλAR(λ,A)vdx=ɛ(λ)(v,vt).
Hence by (4.24), (4.23) is rewritten as
(4.25)ɛ(λ)(v,vt)≤c3‖Δu‖pp+‖u‖qq+p4(t,w).
Note that the right-hand side of (4.25) is independent of λ. So taking limit on both side of (4.25) for λ→∞, association with (2.15), we deduce that
(4.26)12ddt‖∇v‖22≤c3‖Δu‖pp+‖u‖qq+p4(t,w).
Integrating (4.26) from s to t (-1≤s≤t≤0), it yields that
(4.27)‖∇v(t)‖22≤2c3∫st‖Δu(τ)‖ppdτ+2∫st‖u(τ)‖qqdτ+2∫stp4(τ,w)dτ+‖∇v(s)‖22≤2c3∫-10‖Δu(τ)‖ppdτ+2∫-10‖u(τ)‖qqdτ+2∫-10p4(τ,w)dτ+‖∇v(s)‖22.
Therefore, by Lemma 4.1, we find that
(4.28)‖∇v(t)‖22≤2(c3+1)r22(w)+2∫-10p4(τ,w)dτ+‖∇v(s)‖22.
Integrating (4.28) for s from −1 to 0, we have
(4.29)‖∇v(t)‖22≤2(c3+1)r22(w)+2∫-10p4(τ,w)dτ+∫-10‖∇v(s)‖22ds,
for all t∈[-1,0]. By Poincare's inequality, and Young's inequality, there exist positive constants c4, c5, c6 such that
(4.30)‖∇v(s)‖22≤c4‖Δv(s)‖22≤2c4‖Δu(s)‖22+2c4‖Δz(s)‖22≤2c5‖Δu(s)‖pp+2c5‖Δz(s)‖22+c6.
Hence, by (4.30) and using Lemma 4.1 again, (4.29) follows
(4.31)‖∇v(t)‖22≤2(c3+c5+1)r22(w)+2∫-10p4(τ,w)dτ+2c5∫-10‖Δz(τ)‖22dτ+c6,
with t∈[-1,0]. See that v(t)=u(t)-z(t). Then, we have
(4.32)‖∇u(t)‖22≤2‖∇v(t)‖22+2‖∇z(t)‖22≤2(c3+c5+1)r22(w)+2∫-10p4(τ,w)dτ+2c5∫-10‖Δz(τ)‖22dτ+2sup⁡-1≤t≤0‖∇z(t)‖22+c6,
which gives an expression for r32(w). This completes the proof.By Theorem2.2 and Lemma 4.2, we have obtained our main result in this section.Theorem 4.3.
Assume thatg satisfies (1.4)–(1.6) and f is given in V′. Then the RDS φ(t,ω) generated by the stochastic equation (1.1) possesses a random attractor 𝒜(w) defined as
(4.33)A(w)=⋃B∈B(H)⋂s≥0⋃t≥sφ(t,θ-tw)B¯¯,
where ℬ(H) denotes all the bounded subsets of H and the closure is the H-norm.Remark 4.4.
As stated in Theorem3.1, under the assumptions (1.4)–(1.6), the solutions of (1.1) are in W02,p(D). So it is possible in theory to obtain a compact random attractor in W01,p(D) or W02,p(D). But it seems most difficulty to derive the estimate of solution in W2,p(D) due to the nonlinear principle part Δ(Φp(Δu)).


## 5. The Single Point Attractor

In this section, we consider the attracting by a single point. In order to derive our anticipating result, we assume thatC3>0 in (1.6). This leads to the following fact that for every fixed t∈ℝ and w∈Ω, the solution u(t,w;s,u(s)) to (1.1) is a Cauchy sequence in H for the initial time s with initial value u(s) belonging to the bounded subset of H. Then we obtain a compact attractor consisting of a single point which is the limit of u(0,w;s,u(s)) as s→-∞.Lemma 5.1.
Assume thatg satisfies (1.4)–(1.6) and f is given in V′, C3>0. Then for s1≤s2≤t and u01,u02∈H with u(s1)=u01 and u(s2)=u02, there exists a positive constant k<C3 such that
(5.1)‖u(t,w;s1,u01)-u(t,w;s2,u02)‖22≤2e-C3t(4e(C3-k)s2(‖u01‖22+‖z(s1)‖22+∫-∞0p2(τ,w)ekτdτ)+2eks2‖z(s2)‖22+eks2‖u02‖22).
In particular, for each fixed t∈ℝ and w∈Ω there exists a single point ξt(w) in H such that
(5.2)lim⁡s→-∞S(t,s;w)u0=ξt(w),
where u0=u(s) and S(t,s;w) is the stochastic flow defined as in (3.28) which is a version of solution to (1.1). Furthermore, the limit in the above is independent of u0 for all u0 belonging to a bounded subset of H.Proof.
Fors1≤s2≤t and u01,u02∈H with u(s1)=u01 and u(s2)=u02, we can deduce from (3.9) and (3.8) that
(5.3)ddt(u(t,w;s1,u01)-u(t,w;s2,u02))+Ψ¯(u(t,w;s1,u01))-Ψ¯(u(t,w;s2,u02))=0,
where u(t)=v(t)+z(t) is the solution to problem (1.1). On the other hand, by (3.8) and (3.16), we immediately deduce that
(5.4)(Ψ¯(u1)-Ψ¯(u2),u1-u2)≥C3‖u1-u2‖22.
Then, multiplying (5.3) by u(t,w;s1,u01)-u(t,w;s2,u02), integrating over D, and using (5.4), we find that
(5.5)ddt‖u(t,w;s1,u01)-u(t,w;s2,u02)‖22+C3‖u(t,w;s1,u01)-u(t,w;s2,u02)‖22≤0.
Now, applying Gronwall's lemma to (5.5) from s2 to t, it yields that
(5.6)‖u(t,w;s1,u01)-u(t,w;s2,u02)‖22≤‖u(s2,w;s1,u01)-u02‖22e-C3(t-s2)≤2(‖u(s2,w;s1,u01)‖22+‖u02‖22)e-C3(t-s2).
We then estimate ∥u(s2,w;s1,u01)∥22. To this end, we rewrite (3.21) as
(5.7)(Ψ(v),v)≥C12‖u‖qq-p1(t,w).
Since q≥2, by Hölder's inequality and inverse Young's inequality we can choose constant 0<k<C3 such that
(5.8)C12‖u‖qq≥k2‖v‖22-c1‖z‖qq-c2.
Then, by (5.7)-(5.8) it follows from (4.2) that
(5.9)ddt‖v‖22+k‖v‖22≤2p5(t,w),
where p5(t,w)=p1(t,w)+c1∥z∥qq+c2 and v(t)=v(t,w;s1,u01-z(s1)). Using Gronwall's lemma to (5.9) from s1 to s2 with s1≤s2≤0, we get that
(5.10)‖v(s2,w;s1,u01-z(s1))‖22≤‖u01-z(s1)‖22e-k(s2-s1)+∫s1s22p5(τ,w)e-k(s2-τ)dτ≤2e-ks2(‖u01‖22+‖z(s1)‖22+∫-∞0p5(τ,w)ekτdτ).
Similar to the argument of (4.4), we know that the integral in the above is convergent. Therefore, we have
(5.11)‖u(s2,w;s1,u01)‖22≤2‖v(s2,w;s1,u01)‖22+2‖z(s2)‖22≤4e-ks2(‖u01‖22+‖z(s1)‖22+∫-∞0p5(τ,w)ekτdτ)+2‖z(s2)‖22,
from which and (5.6) it follows for every fixed t∈ℝ that
(5.12)‖u(t,w;s1,u01)-u(t,w;s2,u02)‖22≤2e-C3t[4e(C3-k)s2(‖u01‖22+‖z(s1)‖22+∫-∞0p5(τ,w)ekτdτ)+2eks2‖z(s2)‖22+eks2‖u02‖22]⟶0,as,s1,s2⟶-∞,
where the convergence is uniform with respect to u01,u02 belonging to every bounded subset of H. Then (5.12) implies that for fixed t∈ℝ, u(t,w;s,u(s)) is a Cauchy sequence in H for s∈ℝ. Thus, by the completeness of H, for every fixed t∈ℝ and w∈Ω, u(t,w;s,u(s)) has a limit in H denoted by ξt(w), that is,
(5.13)lim⁡s→-∞u(t,w;s,u(s))=ξt(w).Theorem 5.2.
Assume thatg satisfies (1.4)–(1.6) and f is given in V′, C3>0. Then the RDS φ(t,w) generated by the solution to (1.1) possesses a single point attractor 𝒜(w), that is, there exists a single point ξ0(w) in H such that
(5.14)A(w)={ξ0(w)}.Proof.
By Lemma5.1 we define
(5.15)ξ0(w)=lim⁡s→-∞S(0,s;w)u0,
where S(0,s;w)=u(0,w;s,u(s)) by (3.28). Then we need prove that 𝒜(w)={ξ0(w)} is a compact attractor. It is obvious that {ξ0(w)} is a compact random set. Hence by Definition 2.1 it suffices to prove the invariance and attracting property for {ξ0(w)}. Since by the continuity of φ(t,w), and relations (3.29)–(3.32), we have
(5.16)φ(t,w)ξ0(w)=φ(t,w)lim⁡s→-∞S(0,s;w)u0=lim⁡s→-∞φ(t,w)S(0,s;w)u0=lim⁡s→-∞S(t,0;w)S(0,s;w)u0=lim⁡s→-∞S(t,s;w)u0=lim⁡s→-∞S(t-s,0;θsw)u0=lim⁡s→-∞S(0,s-t;θtw)u0=ξ(θtw)
then it follows that φ(t,w)𝒜(w)=𝒜(θtw). On the other hand, by Lemma 5.1, the convergence is uniform with respect to u0 belonging to a bounded subset. Then for every bounded subset B⊂H, by relations (3.32) and (3.28), it follows that
(5.17)dist⁡(φ(t,θ-tw)B,A(w))=sup⁡u0∈B‖φ(t,θ-tw)u0-ξ0(w)‖2=sup⁡u0∈B‖S(0,-t,w)u0-ξ0(w)‖2=sup⁡u0∈B‖u(0,w;-t,u0)-ξ0(w)‖2⟶0
as t→+∞. That is to say 𝒜(w) is a attracting set which attracts every deterministic bounded set of H, and therefore we complete the proof.



---

*Source: 616451-2011-06-30.xml*