DSpace at VNU: Random attractors for stochastic semi-linear degenerate parabolic equations

Contents lists available atScienceDirect Nonlinear Analysis: Real World Applications journal homepage:www.elsevier.com/locate/nonrwa Random attractors for stochastic semi-linear degenera

Contents lists available atScienceDirect Nonlinear Analysis: Real World Applications

journal homepage:www.elsevier.com/locate/nonrwa

Random attractors for stochastic semi-linear degenerate

parabolic equations

Meihua Yanga,∗, P.E Kloedenb

aSchool of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

bInstitute of Mathematics, Goethe University, Frankfurt am Main, 60054, Germany

a r t i c l e i n f o

Article history:

Received 27 January 2011

Accepted 11 April 2011

Keywords:

Stochastic degenerate parabolic equation

Random dynamical system

Random attractor

a b s t r a c t The existence of a random attractor is established for a class of stochastic semi-linear degenerate parabolic equations with the leading term of the form div(σ(x)∇u)and additive spatially distributed temporal noise The nonlinearity is dissipative for large values of the state without restriction on the growth order of the polynomial, while the spatial domain

is either bounded or unbounded

© 2011 Elsevier Ltd All rights reserved

1 Introduction

The theory of nonautonomous attractors for deterministic dissipative systems is now quite well established [1–4] Typically, such systems are generated by a parabolic or hyperbolic evolution equation with a nondegenerate leading term The degenerate case is technically more complicated, and much less has been done, even in the autonomous context [5–7] The situation is similar for stochastic evolution equations These are intrinsically nonautnomous, and they generate a random dynamical system, for which the appropriate attractor concept is a random attractor [8–14] Recently, Röckner and his coauthors [15–17] made significant progress with stochastic porous media equations In this paper, we consider

a different class of stochastic semi-linear degenerate parabolic equations with a different kind of degeneracy condition, specifically with a leading term of the form div(σ (x)∇u)as well as an additive spatially distributed temporal noise The deterministic version of these equations has been investigated by Anh and his coworkers [5,18,19], who also established the existence of nonautonomous attractors when the forcing terms are nonautonomous Their results provide the functional analytic framework for our work here

We formulate the equations in the next section, and outline the required background material on functional analytical results in Section3and on random dynamical systems in Section4 Then, we formulate our main result on the existence of

a random attractor in Section5and, finally, in Section6, establish the required dissipativity estimates on the solutions

2 The equations

We investigate the following stochastic semi-linear degenerate parabolic equation with variable, nonnegative

coefficients defined on an arbitrary domain (bounded or unbounded) D N ⊂RN with N ≥2:







du+ [−div(σ (x)∇u) + λu+f(u)]dt=

m

−

j= 1

h jdwj, x∈D N,t≥0,

u(x,0) =u0(x), x∈D N,

u(x,t)|∂DN =0, t≥0,

(2.1)

∗Corresponding author Tel.: +86 18971660733; fax: +49 0 69 798 28846.

E-mail addresses:yangmeih@gmail.com (M Yang), kloeden@math.uni-frankfurt.de (P.E Kloeden).

1468-1218/$ – see front matter © 2011 Elsevier Ltd All rights reserved.

whereλ >0, and the nonlinear term f ∈C1(R,R)satisfies the following assumptions:

with positive constant l.

(1) The case when D Nis bounded:

with positive constantsα1, α2, β1andβ2

(2) The case when D Nis unbounded:

f(x,s)s≥ α1|s|p−k1(x), |f(x,s)| ≤ α2|s|p−1+k2(x), (2.4) with positive constantsα1andα2, where k1∈L1(D N) ∩ L∞(D N) and k2∈L2(D N) ∩L q(D N), with1p+1

q =1

The degeneracy of problem(2.1)is considered in the sense that the measurable, nonnegative diffusion coefficientσ (x)

is allowed to have at most a finite number of essential zeros As in [18], we assume that the functionσ :D N →R+∪ {0}

satisfies the following assumptions:

(Hα) σ ∈ L1

loc(D N)and, for someα ∈ (0,2), lim infx→z|x−z|− ασ(x) > 0 for every z ∈ D N , when the domain D Nis bounded;

(Hα,β∞) σsatisfies conditionHαand lim inf|x|→∞|x|−βσ(x) >0 for someβ >2, when the domain D Nis unbounded The assumptions(Hα)and(Hα,β)imply, first, that the set of zeros is finite and, second, that the functionσcould be

nonsmooth, e.g., cannot be of class C2ifα ∈ (0,2)and cannot have bounded derivatives ifα ∈ (0,1) See [5–7,18,19] for more details

3 Function spaces and operators

We recall from [5,7,18] some basic concepts and properties about function spaces that we will use later

Let N ≥2 andα ∈ (0,2), and define

2∗α=







4

α ∈ (2, ∞), if N=2,

2N

N−2+ α ∈



N−2

 , if N≥3.

(3.1)

The exponent 2∗

αhas the role of the critical exponent in the classical Sobolev embeddings The following lemma is the

generalized version of a Poincaré inequality [20, Corollary 2.6]; see also [6,7,18]

Lemma 3.1 Let D N be a bounded (respectively, unbounded) domain in R N , N≥2, and assume that conditionHα(respectively,

Hα,β∞) is satisfied Then, there exists a constant c>0 such that

∫

DN

|u|2dx≤c

∫

DNσ (x)|∇u|2dx for every u∈C0∞(D N,R). (3.2)

In the case of a bounded domain, inequality(3.2)holds forα ∈ (0,2]; however, the caseα =2 can be considered as

a ‘‘critical case’’ Moreover, conditionHαis optimal in the sense that forα >2 there exist functions such that(3.2)is not satisfied (see [7,18,20]) Note that, in the case of an unbounded domain, inequality(3.2)generally does not hold ifβ ≤2 in

Hα,β∞ (see also [7,18,20,21])

The natural energy space for problem(2.1)involves the space D1, 2

0 (D N, σ ) defined as the closure of C0∞with respect to the norm,

‖u‖

D1, 2 (DN,σ ):=

∫

DNσ (x)|∇u|2dx

1

The space D1, 2

0 (D N, σ )is a Hilbert space with respect to the scalar product,

(u, v)σ :=

∫

DN

The following two lemmata refer to the continuous and compact inclusions of D1, 2

0 (D N, σ )[20, Propositions 3.3–3.5]; see also [7,18]

Lemma 3.2 Assume that D N is a bounded domain in R N,N≥2, and thatσsatisfies(Hα) Then, the following embeddings hold:

(i) D1, 2

0 (D N, σ ) ↩→L2∗α(D N)continuously,

(ii) D1, 2

0 (D N, σ ) ↩→L p(D N)compactly if p∈ [1,2∗α).

Lemma 3.3 Assume that D N is an unbounded domain in R N,N≥2, and thatσsatisfies(H∞

α,β) Then, the following embeddings hold:

(i) D1, 2

0 (D N, σ ) ↩→L p(D N)continuously for every p∈ [2∗

β,2∗

α],

(ii) D1, 2

0 (D N, σ ) ↩→L p(D N)compactly if p∈ (2∗

β,2∗

α), where 2∗

β= N−2N2 + β(<2).

Remark 3.4 ([ 7 ]) Note that, as a special case, the embedding D1, 2

0 (D N, σ ) ↩→L2(D N)is compact if either condition(Hα)

or condition(H∞

α,β)holds However, sinceσis not in L∞

loc(D N), there is in general no inclusion relation between the space

D1, 2

0 (D N, σ) ↩→L2(D N) and the standard Sobolev space H1(D N)

Friedrich’s extension theory of [22,23] provides the justification for this natural energy space for system(2.1), which has been used in [6,7,18]

Assuming conditions(Hα)and(H∞

α,β), the operator T:= −div(σ(x)∇)is positive and self-adjoint with the domain of definition,

Dom(T) = u∈D1, 2

0 (D N, σ) : Tu∈L2(D N)

The space Dom(T) is a Hilbert space endowed with the usual graph scalar product Moreover, there exists a complete orthonormal system of eigenvectors(e j, λj)j∈Nsuch that

(e i, e j) = δij and −div(σ (x)∇e j) = λj e j, i,j=1,2, , (3.5)

0< λ1≤ λ2≤ · · · , λj → +∞ as j→ ∞

Noting that

λ1=inf







‖u‖2

D1, 2 (DN,σ )

‖u‖2

L2 (DN)

:u∈D1, 2

0 (D N, σ ), u̸=0





 ,

we have

‖u‖2

D1, 2 (DN,σ )≥ λ1‖u‖2

The fractional powers are defined as follows For every s>0, Ts is an unbounded self-adjoint operator in L2(D N)whose domain Dom(Ts) is a dense subset in L2(D N) The operator Tsis strictly positive and injective Also, Dom(Ts)endowed with scalar product(u, v)Dom (Ts) := (Ts u,Tsv)L2 (DN)is a Hilbert space We write V 2s = Dom(Ts)and have the following identifications: Dom



T−1



= D−01(D N, σ ), the dual of D1, 2

0 (D N, σ ), as well as Dom(T0) = L2(D N) and Dom(T1) =

D1, 2

0 (D N, σ) Moreover, the injection V 2s1 ⊂V 2s2for s1>s2is compact and dense

Furthermore, we define D p(T) := u∈D1, 2

0 (D N, σ ) :Tu∈L p(D N) 

4 Random dynamical systems

We recall some basic concepts related to random attractors for stochastic dynamical systems; for more details, see [2,8,

10,11,24]

Let(X, ‖ · ‖X )be a separable Hilbert space with Borelσ-algebraB(X), and let(Ω,F,P)be a probability space

Definition 4.1. (Ω,F,P, (θt )t∈R)is called a metric dynamical system ifθ :R×Ω →Ωis(B(R) ×F,F)-measurable,

θ0is the identity onΩ, θs+t = θt ◦ θsfor all s,t∈R, andθt (P) =P for all t∈R

Definition 4.2 A random dynamical system (RDS)(θ, φ)consists of a metric dynamical system(Ω,F,P, (θt)t∈ R)and a cocycle mappingφ : R+×Ω×X → X , which is(B(R+) ×F ×B(X),B(X))-measurable and satisfies the following properties:

1 φ(0, ω,x) =x (initial condition),

2 φ(s, θtω, φ(t, ω,x)) = φ(s+t, ω,x)(cocycle property),

for all s,t ∈R+, x∈X , andω ∈Ω

It is called a continuous RDS ifφis continuous with respect to x for each t ≥0 andω ∈Ω

To study the asymptotic behavior of the RDS determined by Eq.(2.1), we first need to recall some concepts and properties.

An(F,B(R))measurable function R:Ω →R is said to be tempered with respect to(θt )t∈Rif

lim

t→∞e−γt R(θ−tω) =0

for P-almost allω ∈Ωand allγ >0

A set-valued mapB : Ω → 2X is called a random closed set if B(ω)is nonempty and closed for eachω ∈Ω and the mappingω →d(x,B(ω))is(F,B(R))measurable for all x∈X

A random bounded setB:= {B(ω)}ω∈ Ω of X is said to be tempered with respect to(θt)t∈ Rif R(ω) := diam(B(ω)) :=

supx,y∈B d(x,y)is a tempered random variable

Let D be the collection of all tempered random sets in X We will only consider the class D of tempered random sets in

this paper

Definition 4.3 A random setA := {A(ω)}ω∈ Ω ∈ X is called a D-random attractor (or D-pullback attractor) for an RDS (θ, φ)on a Polish space(X,d)if

1 Ais a random compact set, i.e., A(ω)is nonempty and compact for P-almost allω ∈Ωandω →d(x,A(ω))is measurable

for every x∈X ;

2 A is aφ-invariant, i.e.,φ(t, ω,A(ω)) =A(θt ω), for all t≥0 and P-almost allω ∈Ω;

3 A attracts all tempered random setsD= {D(ω)}ω∈ Ω∈Din the sense that

lim

t→∞dist(φ(t, θ−tω, D(θ−tω)),A(ω)) =0, P-almost allω ∈Ω.

When the RDS is continuous, the existence of a random compact absorbing set is a sufficient condition for the existence

of a random attractor [9–11,13,25]

Definition 4.4 A random closed setB = {B(ω)}ω∈ Ω ∈ Dis called a D-random absorbing set for an RDS(θ, φ)if, for any

D = {D(ω)}ω∈ Ω∈Dand P-almost allω ∈Ω, there exists a TD(ω) >0 such that

Theorem 4.5 Let (θ, φ)be a continuous RDS on a Polish space(X,d) Suppose that there exists a closed random absorbing set B = {B(ω)}ω∈ Ω ∈ Dand that φis D-pullback asymptotically compact in X Then,φ has a unique D-random attractor

A = {A(ω)}ω∈ Ω, which is unique in the class of tempered random sets, with component subsets given by

A(ω) = 

τ≥ 0



t≥ τ

φ(t, θ−tω, B(θ−tω)), ω ∈Ω.

5 Existence of a D-pullback attractor

In this paper, we will prove that the stochastic degenerate parabolic equation in D N generates a continuous random

dynamical system, which has a D-random attractor in L2(D N)







du+ [Au+ λu+f(u)]dt=

m

−

j= 1

h jdwj, x∈D N,t≥0

u(x,0) =u0(x), x∈D N,

u(x,t)|∂DN =0, t ≥0,

(5.1)

where Au:= −div(σ (x)∇u)and h1, ,h m∈L p(D N)∩Dom(A)∩D p(A), for some p≥2 (given in(2.2)) and thew1, , wm

are pairwise independent two-sided real-valued Wiener processes on the following probability space

We consider the canonical probability space(Ω,F,P), where

Ω =  ω = (ω1, ω2, , ωm) ∈C(R,Rm) : ω(0) =0

andF is Borelσ-algebra induced by the compact open topology ofΩ, while P is the corresponding Wiener measure on

(Ω,F) Then, we identifyωwith

W(t, ω) = (w1(t), w2(t), , wm(t)) = ω(t) for t∈R.

Finally, define the time shift by

θt ω(·) = ω(· +t) − ω(t), ω ∈Ω,t ∈R.

Then,(Ω,F,P, (θt )t )is a metric dynamical system

We now associate a continuous random dynamical system with stochastic degenerate parabolic equation over

(Ω,F,P, (θt )t∈R) To this end, we need to convert the stochastic equation with an additive noise term into a pathwise random partial differential equation

Given j=1, ,m, consider the stochastic stationary solution of the one-dimensional Ornstein–Uhlenbeck equation

For this solution, the random variable|z j(ωj)| is tempered, and z j((θtω)j)is P-almost sure continuous Therefore, there exists

a tempered function r(ω) >0 such that

m

−

j= 1

where r(ω)satisfies, P-almost surely,

r(θt ω) ≤eλ2 |t|

Then, it follows from(5.3)–(5.4)that, P-almost surely,

m

−

j= 1

|z j((θtω)j)|2+ |z j((θtω)j)|p ≤eλ2 |t|

Putting z(θtω) := ∑m

j= 1h j z j((θtω)j), by(5.2), we see that

dz+ λz dt=

m

−

j= 1

h jdwj.

The existence of a solution to the stochastic degenerate parabolic equation(5.1)follows from [26] We show that problem

(5.1)generates a random dynamical system Letv(t) :=u(t) −z(θtω), where u is a solution of(5.1) Then,vsatisfies

∂v

We know from [18] that−A is the infinitesimal generator of a C0-semigroup e−At on D1, 2

0 (D N, σ ), and it is easy to check from assumptions(2.2)–(2.4)that the function f(·, ω) : D1, 2

0 (D N, σ ) → D1, 2

0 (D N, σ )is locally Lipschitz continuous Thus, the classical semigroup theory on the local existence and uniqueness of the solutions of evolution differential equations

in [27] applies here pathwise Then, by the Galerkin method, one can show that, for allv0∈L2(D N),P-almost surely,(5.6)

has a unique and continuous solutionv(·, ω, v0) ∈C([0, ∞),L2(D N)) ∩L2((0,T);D1, 2

0 (D N, σ )) ∩L p((0,T);L p(D N))with the initial valuev(0, ω, v0) = v0in L2(D N)for every T≥0

Let u(t, ω,u0) = v(t, ω,u0−z(ω)) +z(θt ω) Then, the process u is the solution of problem(5.1) We now define a mappingφ :R+×Ω×L2(D N) →L2(D N)by

for all(t, ω,u0) ∈R+×Ω ×L2(D N) Then,φsatisfies the conditions inDefinition 4.2and, hence,(θ, φ)is a continuous random dynamical system associated with stochastic degenerate parabolic equation(5.1)

In the next section, we establish uniform estimates for the solutions of(5.1)needed to prove the following theorem on the existence of a random attractor for(θ, φ)

Theorem 5.1 Assume that(2.2)–(2.4)hold Then, the continuous random dynamical system(θ, φ)generated by(5.1)has a unique D-random attractor in L2(D N) belonging to the class D.

Proof First, notice thatφhas a closed random absorbing setB = {B(ω)}ω∈ Ω in D byLemma 6.1 Thanks to the compact embedding (seeRemark 3.4),φis D-pullback compact in L2(D N)byLemma 6.5 Hence, the existence of a unique D-random attractor for the RDS(θ, φ)belonging to the class D follows fromTheorem 4.5 

Remark 5.2 In the case of an unbounded domain D N ⊂ RN, N ≥ 2, we assume thatσ satisfies the condition(Hα,β∞)

Then, the operator A:= −div(σ(x)∇)has the same properties as in the case of a bounded domain Therefore, one can apply

similar methods as for the bounded case, with some small changes in the conditions on f More precisely, we assume that

f :D N×R→R is a Carathéodory function and that it satisfies the following assumptions: for all x∈RN and s∈R,

∂f

∂s(x,s) ≥ −l,





∂f

∂s(x,s)



 ≤ ϕ1(x)

f(x,s)s≥ α1|s|p− ϕ2(x), |f(x,s)| ≤ α2|s|p−1+ ϕ3(x),

with positive constantsα1, α2and l, whereϕ1∈L2(RN), ϕ2∈L1(RN) ∩L∞(RN), andϕ3∈L2(RN) ∩L q(RN), with1p+1

q =1 See [9,18] for more details

6 Uniform estimates of solutions

In this section, we establish uniform estimates for the solutions of(5.1)by the uniform estimates for the solutions of(5.6)

Henceforth, we always assume that D is the collection of all tempered subsets of L2(D N)with respect to(Ω,F,P, (θt )t∈ R) The next lemma shows that the RDS(θ, φ)has a random absorbing set in D

Lemma 6.1 Assume that(2.2)–(2.4)hold Then, there exists a random absorbing setB= {B(ω)}ω Ω ∈Dfor the RDS(θ, φ).

Proof We first derive uniform estimates onv(t) = u(t) −z(θt ω)from which the uniform estimates on u(t)follow immediately

Multiplying(5.6)byvand then integrating over D N, we find that

1

2

d

dt

∫

DN

| v|2dx+

∫

DN

σ (x)|∇v|2dx+

∫

DN

λ|v|2dx= −

∫

DN

f(v +z(θtω))vdx−

∫

DN

For the nonlinear term, by(2.3)and(2.4), we obtain

∫

DN

f(v +z(θtω))vdx=

∫

DN

f(v +z(θt ω))(v +z(θt ω))dx−

∫

DN

f(v +z(θt ω))z(θt ω)dx

≥ α1

∫

DN

|u|p dx− β1mes D N−

∫

DN

f(v +z(θt ω))z(θtω)dx

≥ α1

∫

DN

|u|p dx−

∫

DN

|u|p−1|z(θt ω)|dx− β1mes D N

≥ α1 2

∫

DN

|u|p dx−c1

∫

DN

On the other hand,





∫

DN

Az(θt ω)vdx



 ≤ λ

2

∫

DN

| v|2dx+ 1

2λ

∫

DN

and it follows from(6.1)–(6.3)that

d

dt

∫

DN

| v|2dx+ λ ∫

DN

| v|2dx+

∫

DN

σ (x)|∇v|2dx+ α1

∫

DN

|u|p dx

≤ c2

∫

DN

|Az(θt ω)|2

∫

DN

|z(θtω)|p

dx



Note that z(θt ω) = ∑m

j= 1h j z j((θtω)j) and h j∈L p(D N) ∩Dom(A) ∩D p(A) Therefore, let

p1(θtω) =c2

m

−

j= 1

Then, it follows from(5.5)that, P-almost surely,

Hence, for all t≥0,

d

dt

∫

DN

| v|2dx+ λ ∫

DN

| v|2dx+

∫

DN

σ (x)|∇v|2dx+ α1

∫

DN

Applying Gronwall’s lemma, we find that, for all t ≥0,

‖ v(t, ω, v0(ω))‖2

L2 (DN)≤e−λt‖ v0(ω)‖2

L2 (DN)+

∫ t

eλ(τ−t)p

1(θτω)dτ +c3

Then, replacingωbyθ−tω , for all t≥0, we obtain

‖ v(t, θ−tω, v0(θ−tω))‖2

L2 (DN)≤ e−λt‖ v0(θ−tω)‖2

L2 (DN)+

∫ t

0

eλ(s−t)p

1(θs−tω)ds+c3

λ

≤ e−λt‖ v0(θ−tω)‖2

L2 (DN)+

∫ 0

−t

eλτp

1(θτω)dτ +c3

λ

≤ e−λt‖ v0(θ−tω)‖2

L2 (DN)+c4

∫ 0

−t

e1λτr(ω)dτ + c3

λ

≤ e−λt‖ v0(θ−tω)‖2

L2 (DN)+

2c4

λ r(ω) +

c3

Recall thatφ(t, ω,u0(ω)) = v(t, ω,u0(ω) −z(ω)) +z(θtω) Hence, the above estimate(6.9)implies, for all t≥0, that

‖ φ(t, θ−tω, u0(θ−tω))‖2

L2 (DN) ≤2‖ v(t, θ−tω,u0(θ−tω))‖2

L2 (DN)+2‖z(ω)‖2

L2 (DN)

≤2e−λt‖u0(θ−tω) −z(θ−tω)‖2

L2 (DN)+c5r(ω) +c3

λ +2‖z(ω)‖2L2 (DN)

≤4e−λt ‖u0(θ−tω)‖2

L2 (DN)+ ‖z(θ−tω)‖2

L2 (DN)

 +c5r(ω) +c3

λ +2‖z(ω)‖2

By assumption,D= {D(ω)}ω∈ Ω ∈Dis tempered On the other hand, by definition,‖z(ω)‖2

L2 (DN)is also tempered Therefore,

if u0(θ−tω) ∈D(θ−tω) , then there exists a TD(ω) >0 such that, for all t≥TD(ω),

4e−λt ‖u0(θ−tω)‖2

L2 (DN)+ ‖z(θ−tω)‖2

L2 (DN)



≤c6r(ω) +c6,

which, along with(6.10), shows that, for all t≥TD(ω),

‖ φ(t, θ−tω, u0(θ−tω))‖2

L2 (DN)≤2



c6r(ω) +c6+ ‖z(ω)‖2

Givenω ∈Ω, define

B(ω) = 

u∈L2(D N) : ‖u‖2

L2 (DN)≤2



c6r(ω) +c6+ ‖z(ω)‖2

Then,B= {B(ω)}ω∈ Ω ∈D Further,(6.11)implies thatBis a random absorbing set for the RDS(θ, φ)in D 

We next derive uniform estimates forvin D1, 2

0 (D N, σ )and for u in L p(D N)

Lemma 6.2 Assume that(2.2)–(2.4)hold LetD = {D(ω)}ω∈ Ω ∈Dand u0(ω) ∈D(ω) Then, for every T1≥0, the solution

u(t, ω,u0(ω))of problem(5.1)and the solutionv(t, ω, v0(ω))of problem(5.6)withv0(ω) =u0(ω) −z(ω)satisfy, P-almost surely,

∫ t

T1

eλ(s−t)‖u(s, θ−tω, u0(θ−tω))‖p

L p(DN)ds≤e−λt‖ v0(θ−tω)‖2

∫ t

T1

eλ(s−t)‖ v(s, θ−tω, v0(θ−tω))‖2

D1, 2 (DN,σ )ds≤e

− λt‖ v0(θ−tω)‖2

for t≥T1.

Proof First, replacing t by T1and then replacingωbyθ−tωin(6.8), the following inequality holds:

‖ v(T1, θ−tω, v0(θ−tω))‖2

L2 (DN)≤e

− λT1‖ v0(θ−tω)‖2

L2 (DN)+

∫ T1

0

eλ(s−T1 )p

1(θs−tω) ds+c.

Multiplying by eλ 1 (T1 −t)gives

eλ(T1 −t)‖ v(T1, θ−tω, v0(θ−tω))‖2

L2 (DN)≤e−λt‖ v0(θ−tω)‖2

L2 (DN)+

∫ T1

0

eλ(s−t)p

1(θs−tω) ds+ceλ(T1 −t) (6.15)

where, from(6.6),

∫ T1

eλ(s−t)p

1(θs−tω)ds=

∫ T1 −t

eλτp

It follows that

eλ(T1 −t)‖ v(T1, θ−tω, v0(θ−tω))‖2

L2 (DN)≤e−λt‖ v0(θ−tω)‖2

L2 (DN)+

2

λc4r(ω)e1λ(T1 −t)+ceλ(T1 −t). (6.17) Note that, from(6.7), for t≥T1,

2‖ v(t, ω, v0(ω))‖2

L2 (DN)+

∫ t

T1

eλ(s−t) ‖ v(s, ω, v0(ω))‖2

D1, 2 (DN,σ)+2α1‖u(s, ω,u0(ω))‖p

L p(DN)



ds

≤ 2eλ(T1 −t)‖ v(T1, ω, v0(ω))‖2

L2 (DN)+2

∫ t

T1

eλ(s−t)p

1(θsω)ds+2c3

∫ t

T1

Replacingωbyθ−tω, we obtain, for all t≥T1,

∫ t

T1

eλ(s−t)‖ v(s, θ−tω, v0(θ−tω))‖2

D1, 2 (DN,σ )ds+2α1

∫ t

T1

eλ(s−t)‖u(s, θ−tω,u0(θ−tω))‖p

L p(DN)ds

≤ 2eλ(T1 −t)‖ v(T1, θ−tω, v0(θ−tω))‖2

L2 (DN)+2

∫ t

T1

eλ(s−t)p

1(θs−tω)ds+2c3

∫ t

T1

eλ(s−t)ds

≤ 2eλ(T1 −t)‖ v(T1, θ−tω, v0(θ−tω))‖2

L2 (DN)+2

∫ 0

T1 −t

eλτp

where

2

∫ 0

T1 −t

eλτp

1(θτω)dτ ≤c4r(ω)

∫ 0

T1 −t

Then, it follows from(6.17)and(6.19)that

∫ t

T1

eλ(s−t) ‖ v(s, θ−tω, v0(θ−tω))‖2

D1, 2 (DN,σ )+2α1‖u(s, θ−tω,u0(θ−tω))‖p

L p(DN)



ds

≤ e−λt‖ v0(θ−tω)‖2

As a special case ofLemma 6.2, we have the following estimates

Lemma 6.3 Assume that (2.2)–(2.4)hold LetD = {D(ω)}ω∈ Ω ∈ Dand u0(ω) ∈ D(ω) Then, P-almost surely, there exists

a TD(ω) > 0 such that the solution u(t, ω,u0(ω))of problem(5.1)and the solutionv(t, ω, v0(ω))of problem(5.6)with

v0(ω) =u0(ω) −z(ω)satisfy

∫ t+ 1

t

‖u(s, θ−t− 1ω,u0(θ−t− 1ω))‖p

∫ t+ 1

t

‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

for all t ≥TD(ω).

Proof First, replacing t by t+1 and T1by t in(6.14), we obtain that

∫ t+ 1

t

eλ 1 (s−t− 1 )‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ )ds≤e

− λ 1 (t+ 1 )‖ v0(θ−t− 1ω)‖2

L2 (DN)+c(1+r(ω)). (6.24) Then, noting that eλ 1 (s−t− 1 )≥e− λ 1, for s∈ [t,t+1],

e−λ 1

∫ t+ 1

t

‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ )ds≤e

− λ 1 (t+ 1 )‖ v0(θ−t− 1ω)‖2

L2 (DN)+c(1+r(ω))

≤2e−λ 1 (t+ 1 )



‖u0(θ−t− 1ω)‖2

L2 (DN)+ ‖z(θ−t− 1ω)‖2

L2 (DN)



Since‖u0(ω)‖2

L2 (DN)and‖z(ω)‖2

L2 (DN)are tempered, there is a TD(ω) >0 such that 2e−λ 1 (t+ 1 )(‖u0(θ−t− 1ω)‖2

2 ( )+ ‖z(θ−t− 1ω)‖2

for all t≥TD(ω), which, with(6.25), implies that

∫ t+ 1

t

‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ )ds≤2e

for all t≥TD(ω)

Similarly, we find that

∫ t+ 1

t

‖u(s, θ−t− 1ω,u0(θ−t− 1ω))‖p

for all t≥TD(ω) 

Lemma 6.4 Assume that(2.2)–(2.4)hold LetD = {D(ω)}ω∈ Ω ∈ Dand u0(ω) ∈D(ω) Then, P-almost surely, there exists a

TD(ω) >0 such that the solution u(t, ω,u0(ω))of problem(5.1)satisfies

∫ t+ 1

t

‖u(s, θ−t− 1ω,u0(θ−t− 1ω))‖2

for all t≥TD(ω).

Proof Let TD(ω)be as given inLemma 6.3, and take t≥TD(ω)and s∈ (t,t+1) We obtain that

‖u(s, θ−t− 1ω,u0(θ−t− 1ω))‖2

D1, 2 (DN,σ ) ≤2‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ)+2‖z(θs−t− 1ω)‖2

D1, 2 (DN,σ), (6.30) where

‖z(θs−t− 1ω)‖2

D1, 2 (DN,σ )≤c

m

−

j= 1

|z j((θs−t− 1ω)j)|2≤ceλ1(t+ 1 −s)r(ω)j ≤ceλ1r(ω). (6.31)

Integrating(6.30)with respect to s over(t,t+1), byLemma 6.3and inequality(6.31), we obtain

∫ t+ 1

t

‖u(s, θ−t− 1ω,u0(θ−t− 1ω))‖2

Then, lemma follows from(6.32) 

Finally, we derive uniform estimates for u in D1, 2

0 (D N, σ)

Lemma 6.5 Assume that(2.2)–(2.4)hold LetD = {D(ω)}ω∈ Ω ∈ Dand u0(ω) ∈D(ω) Then, P-almost surely, there exists a

TD(ω) >0 such that

‖u(t, θ−tω,u0(θ−tω))‖2

for all t≥TD(ω), where c is positive deterministic and r(ω)is the tempered function given in(5.3).

Proof Taking the inner product of(5.6)with Av = −div(σ (x)∇v)in L2(D N), we obtain

1

2

d

dt‖ v‖2

D1, 2 (DN,σ )+

∫

DN

|Av|2dx+ λ

∫

DN

σ(x)|∇v|2dx= −

∫

DN

f(u)Av −

∫

DN

We first estimate the nonlinear term in(6.34) By(2.2)–(2.4), we have

∫

DN

f(u)Avdx=

∫

DN

f(u)Au dx−

∫

DN

f(u)Az(θt ω)dx

=

∫

DN

f′(u)σ (x)|∇u|2dx−

∫

DN

where, from(2.2),

∫

DN

f′(u)σ (x)|∇u|2dx≥ −l

∫

DN

and, from(2.4),



−

∫

f(u)Az(θt ω)dx



 ≤ α3

∫

|u|p−1|Az|dx+c

∫

|u|p dx+c

∫



−

∫

DN

Az(θtω)Avdx



 ≤ 1 2

∫

DN

|Av|2dx+1

2

∫

DN

Then, we obtain

1

2

d

dt‖ v‖2

D1, 2

(DN,σ )+

∫

DN

|Av|2dx≤l

∫

DN

σ (x)|∇u|2dx+c

∫

DN

|u|p dx

+c

∫

DN

|Az(θtω)|p dx+

∫

DN

|Az(θtω)|2dx



Let

p2(θtω) =c

∫

DN

|Az(θt ω)|p

dx+

∫

DN

|Az(θtω)|2

dx



Since z(θtω) = ∑m

j= 1h j z j((θtω)j)and h j∈L p(D N) ∩Dom(A) ∩D p(A), there are positive constants k1and k2such that

p2(θtω) ≤k1

m

−

j= 1

which shows that

p2(θtω) ≤k1eλ2 |t|

Hence,

d

dt‖ v‖2

D1, 2 (DN,σ )≤c



‖u‖2

D1, 2 (DN,σ )+ ‖u‖

p

L p(DN)



Let TD(ω)be the positive constant inLemma 6.3 Take t≥TD(ω)and s∈ (t,t+1), and then integrate(6.43)over(s,t+1)

to obtain

‖ v(t+1, ω, v0(ω))‖2

D1, 2 (DN,σ)≤ ‖ v(s, ω, v0(ω))‖2

D1, 2 (DN,σ)+

∫ t+ 1

s

p2(θτω)dτ

+c

∫ t+ 1

s



‖u(τ, ω,u0(ω))‖2

D1, 2 (DN,σ)+ ‖u(τ, ω,u0(ω))‖p

L p(DN)



dτ

≤ ‖ v(s, ω, v0(ω))‖2

D1, 2 (DN,σ)+

∫ t+ 1

t

p2(θτω)dτ

+c

∫ t+ 1

t



‖u(τ, ω,u0(ω))‖2

D1, 2 (DN,σ)+ ‖u(τ, ω,u0(ω))‖p

L p(DN)



Now, integrating the above inequality with respect to s over(t,t+1), we find that

‖ v(t+1, ω, v0(ω))‖2

D1, 2 (DN,σ)≤

∫ t+ 1

t

‖ v(s, ω, v0(ω))‖2

D1, 2 (DN,σ)ds+

∫ t+ 1

t

p2(θτω)dτ

+c

∫ t+ 1

t



‖u(τ, ω,u0(ω))‖2

D1, 2 (DN,σ)+ ‖u(τ, ω,u0(ω))‖p

L p(DN)



Replacingωbyθ−t− 1ω, we obtain

‖ v(t+1, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ )

≤

∫ t+ 1

t

‖ v(s, θ−t− 1ω, v0(θ−t− 1ω))‖2

D1, 2 (DN,σ )ds+

∫ t+ 1

t

p2(θτ−t− 1ω)dτ

+c

∫ t+ 1

(‖u(τ, θ−t− 1ω,u0(θ−t− 1ω))‖2

D1, 2 (DN,σ)+ ‖u(τ, θ−t− 1ω,u0(θ−t− 1ω))‖p