Furthermore, the limiting behavior of random attractors of the random dynamical systems as stochastic perturbations ap- proach zero is studied and the upper semicontinuity is proved

Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 87, pp. 1–27.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

EXISTENCE AND UPPER SEMICONTINUITY OF RANDOM ATTRACTORS FOR STOCHASTIC p-LAPLACIAN EQUATIONS

ON UNBOUNDED DOMAINS

JIA LI, YANGRONG LI, HONGYONG CUI

Abstract. The existence of a pullback attractor is established for a stochastic p-Laplacian equation onRⁿ. Furthermore, the limiting behavior of random attractors of the random dynamical systems as stochastic perturbations ap- proach zero is studied and the upper semicontinuity is proved.

1. Introduction

It is known that p-Laplacian equation is always used to model a variety of phys- ical phenomena. In this paper, we investigate the asymptotic behavior of solutions to the following stochastic p-Laplacian equation with multiplicative noise defined on the entire spaceRⁿ:

du+ (−div(|∇u|^p−2∇u) +λu)dt= (f(x, u) +g(x))dt+εu◦dW(t), (1.1) where ε >0 is a small positive parameter, λ >0,p≥2 are fixed constants. g is a given function defined onRⁿ,f is a smooth nonlinear function satisfying certain conditions, and W is a two-sided real-valued Wiener processes on a probability space which will be specified later.

The long-term behavior of random systems is captured by a pullback random attractor, which was introduced by [8, 9] as an extension of the attractors theory of deterministic systems in [2, 11, 20, 22, 23]. In the case of bounded domains, the existence of random attractors for stochastic PDEs has been studied extensively by many authors (see [1, 8, 9, 17, 18, 19, 26, 29, 32]) and the reference therein. Since sobolev embeddings are not compact on unbounded domains, it is more difficult to discuss the existence of random attractors for PDEs defined on unbounded do- mains. Nevertheless, the existence of such attractors for some stochastic PDEs on unbounded domains has been proved in [3, 10, 24, 25, 27, 28, 31].

The first aim of this paper is to investigate the existence of random attractors for the stochastic p-Laplacian equation (1.1) defined on Rⁿ. We mention that the existence of global attractors for the p-Laplacian equation in the deterministic case has been discussed by many authors, for examples, in [30, 6] for bounded domains and in [15, 16] for unbounded domains. Recently [29] investigate the

2000Mathematics Subject Classification. 37L30, 35B40, 35B41, 35K55, 35K57, 35Q80.

Key words and phrases. Stochastic p-Laplacian equations; multiplicative noise;

random dynamical system, random attractor; upper semicontinuity.

c

2014 Texas State University - San Marcos.

Submitted January 19, 2014. Published April 2, 2014.

1

existence of random attractors for the p-Laplacian equation with multiplicative noise. However, in the paper [29], the p-Laplace equation is defined in a bounded domain where compactness of Sobolev embeddings is available. To overcome the difficulty caused by the non-compactness of Sobolev embedding onRⁿ, we use the tail-estimate method which is always used to deal with the problem caused by the unboundedness of domains (see [3, 24, 25, 27, 28]). So far as we know, there were no results on random attractors for stochastic p-Laplacian equation with multiplicative noise on unbounded domains.

The second aim of this paper is to examine the limiting behavior of random attractors when ε → 0 and prove the upper semicontinuity of these perturbed random attractors. It is worth mentioning that such continuity of attractors has been investigated, for examples, in [12, 13, 14, 23] for deterministic equations, in [4, 5, 21, 29] for stochastic PDEs in bounded domains and in [10, 28] for stochastic PDEs on unbounded domains.

The paper is arranged as follows. In the next section, we review the pullback random attractors theory for random dynamical systems. In section 3, we define a continuous random dynamical system for the stochastic p-Laplacian equation on Rⁿ. Section 4 is devoted to obtaining uniform estimates of solutions as t → ∞.

These estimate are necessary for proving the existence of bounded absorbing sets and the asymptotic compactness of the solution operator. Finally, we prove the upper semicontinuity of random attractors for (1.1) in the last section.

We denote byk·kand (·,·) the norm and the inner product inL²(Rⁿ) respectively andk · k_p to denote the norm inL^p(Rⁿ). Otherwise, the norm of a general Banach spaceX is written ask · kX.

The lettersc andC(ω) are generic positive constants and positive random vari- able respectively, which don’t depend onε and may change their values from line to line or even in the same line.

2. Preliminaries

In this section, we recall some basic concepts related to RDS (see [1, 7, 8, 9, 28]

for details).

Let (X,k · k_X) be a Banach space with Borelσ-algebra.

Definition 2.1. Let (Ω,F,P) be a probability space and{θt : Ω→Ω, t ∈R} a family of measure preserving transformations such that (t, ω)7→θtωis measurable, θ0=idandθt+s=θtθsfor alls, t∈R. The flowθttogether with the corresponding probability space (Ω,F,P, θt) is called a measurable dynamical system.

Definition 2.2. A continuous random dynamical system(RDS) on X over θ on (Ω,F,P) is a measurable map

ϕ:R⁺×Ω×X 7→X, (t, ω, x)7→ϕ(t, ω)x such that P-a.s.

(i) ϕ(0, ω) =idonX;

(ii) ϕ(t+s, ω) =ϕ(t, θsω)ϕ(s, ω) for all s, t∈R⁺ (cocycle property);

(iii) ϕ(t, ω) :X7→X is continuous.

Definition 2.3. A random compact set {K(ω)}ω∈Ω is a family of compact sets indexed byωsuch that for everyx∈X the mappingω7→d(x, K(ω)) is measurable

with respect toF. A random set {K(ω)}ω∈Ω is said to be bounded if there exist u0∈X and a random variableR(ω)>0 such that

K(ω)⊂ {u∈X,ku−u0kX ≤R(ω)} for allω∈Ω.

Definition 2.4. A random bounded set{B(ω)}_ω∈Ωof X is called tempered with respect to (θt)_t∈Rif for P-a.e.ω∈Ω,

t→∞lim e^−βtd(B(θ_−tω)) = 0 for allβ >0, whered(B) = sup_x∈Bkxk_X.

Definition 2.5. LetDbe a collection of random subsets of X and{K(ω)}ω∈Ω∈ D.

Then {K(ω)}ω∈Ω is called a random absorbing set forϕ in Dif for every B ∈ D and P-a.e. ω∈Ω, there exists tB(ω)>0 such that

ϕ(t, θ−tω, B(θ−tω))⊆K(ω) for allt≥tB(ω).

Definition 2.6. LetDbe a collection of random subsets of X. Thenϕis said to be D-pullback asymptotically compact inXif for P-a.e. ω∈Ω,{ϕ(tn, θ_−tnω, xn)}^∞_n=1 has a convergent subsequence in X whenever tn → ∞, and xn ∈ B(θ_−tω) with {B(ω)}_ω∈Ω∈ D.

Definition 2.7. Let D be a collection of random subsets of X. Then a random set{A(ω)}_ω∈Ω ofX is called aD-random attractor(orD-pullback attractor) forϕ if the following conditions are satisfied, for P-a.e.ω∈Ω,

(i) A(ω) is a random compact set;

(ii) A(ω) is invariant, that is,ϕ(t, ω,A(ω))=A(θtω), for allt≥0;

(iii) A(ω) attracts every set inD, that is, for everyB ={B(ω)}ω∈Ω∈ D,

t→∞lim d(ϕ(t, θ_−tω, B(θ_−tω)),A(ω)) = 0, wheredis the Hausdorff semi-metric.

Proposition 2.8. Let D be an inclusion-closed collection of random subsets ofX and ϕ a continuous RDS on X over (Ω,F,P,(θt)t∈R). Suppose that {K(ω)}ω∈Ω

is a closed random absorbing set for ϕ in D and ϕ is D-pullback asymptotically compact inX. Thenϕhas a uniqueD-random attractor{A(ω)}ω∈Ωwhich is given by

A(ω) =∩T≥0∪_t≥Tϕ(t, θ_−tω, K(θ_−tω)).

Proposition 2.9. Let D be an inclusion-closed collection of random subsets of X. Givenσ >0, suppose ϕ_σ is a random dynamical system over a metric system (Ω,F,P,(θ_t)_t∈R) which has a D-random attractor A_σ and ϕ₀ is a deterministic dynamical system defined on X which has a global attractor A₀. Assume that the following conditions be satisfied: (i) For P-a.e. ω ∈ Ω, t ≥0, σn → 0, and xn, x∈X withxn→x, there holds

n→∞lim ϕ_σn(t, ω, x_n) =ϕ₀(t)x. (2.1) (ii) Every ϕσ has a random absorbing set Eσ ={Eσ(ω)}ω∈Ω∈ D such that for some deterministic positive constantc and forP-a.e. ω∈Ω,

lim sup

σ→0

kEσ(ω)kX ≤c, (2.2)

wherekEσ(ω)kX= sup_x∈Eσ(ω)kxkX.

(iii) There existsσ0>0 such that for P-a.e. ω∈Ω,

∪_0<σ≤σ0A_σ(ω) is precompact in X. (2.3) Then for P-a.e. ω∈Ω,

dist(A_σ(ω),A₀)→0, asσ→0. (2.4) 3. Stochastic p-Laplacian equation with multiplicative noise Here we show that there is a continuous random dynamical system generated by the stochastic p-Laplacian equation defined onRⁿ with multiplicative noise:

du+ (−div(|∇u|^p−2∇u) +λu)dt= (f(x, u) +g(x))dt+εu◦dW(t), (3.1) forx∈Rⁿ,t >0, with the initial condition

u(x,0) =u0(x), x∈Rⁿ. (3.2)

whereε >0,λ >0,p≥2 are constants,g∈L²(Rⁿ),W is a two-sided real-valued Wiener processes on a probability space which will be specified below, andf is a smooth nonlinear function satisfying the following conditions: For allx∈Rⁿ and s∈Rⁿ,

f(x, s)s≤ −α1|s|^p+ψ1(x), (3.3)

|f(x, s)| ≤α2|s|^p−1+ψ2(x), (3.4) whereα1, α2 are positive constants,ψ1∈L¹(Rⁿ)∩L^p2(Rⁿ),ψ2∈L²(Rⁿ)∩L^q(Rⁿ) with ¹_p+¹_q = 1.

In the sequel, we consider the probability space (Ω,F,P) where Ω ={ω∈C(R,R) :ω(0) = 0},

F is the Borel σ-algebra induced by the compact-open topology of Ω, and P the corresponding Wiener measure on (Ω,F). Define the time shift by

θtω(·) =ω(·+t)−ω(t), ω∈Ω, t∈R. Then (Ω,F,P,(θ_t)_t∈R) is a metric dynamical system.

We now associate a continuous random dynamical system with the equation over (Ω,F,P,(θ_t)_t∈R). To this end, we need to convert the stochastic equation with a random multiplicative term into a deterministic equation with a random parameter.

Consider the stationary solutions of the one-dimensional Ornstein-Uhlenbeck equation:

dz+zdt=dW(t). (3.5)

The solution to (3.5) is given by z(θtω) =−

Z 0

−∞

e^τ(θtω)(τ)dτ, t∈Rⁿ. (3.6) From [1, 3, 24, 25], the random variable |z(ω)| is tempered, and there is a θ_t- invariant setΩe ⊂Ω of full P measure such thatz(θtω) is continuous int for every ω∈Ω ande

t→±∞lim

|z(θtω)|

|t| = 0; (3.7)

t→±∞lim 1 t

Z t

0

z(θsω)ds= 0. (3.8)

Following the properties of the O-U process and (3.7),(3.8), it is easy to show for for all 0< ε≤1,

0≤Mε(ω) :=

Z 0

−∞

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds

≤ Z 0

−∞

e^{2|z(θsω)|+2|Rs0z(θτω)dτ|+λs}ds <+∞,

(3.9)

and

0≤K(ω) := max

−2≤τ≤0|z(θ_τ(ω))|<+∞, (3.10) which will be used frequently in the following paper. And it is easy to see that M_ε(ω) andK(ω) are both tempered.

To show that problem (3.1)-(3.2) generates a random dynamical system, let v(t) =e^{−εz(θtω)}u(t), (3.11) whereuis a solution of problem (3.1)-(3.2). Thenv satisfies

dv

dt =e^{ε(p−2)z(θtω)}div(|∇v|^p−2∇v)−λv+e^{−εz(θtω)}(f(x, u) +g(x)) +εvz(θ_tω), (3.12) and the initial condition

v(x,0) =v0(x) =e^−εz(ω)u0(x), x∈Rⁿ. (3.13) Using the standard Galerkin method, one may show that for all v0 ∈L²(Rⁿ), problem (3.12)-(3.13) has a unique solution

v(·, ω, v0)∈C([0,∞), L²(Rⁿ))∩L²((0, T), W^1,p(Rⁿ)).

Furthermore, the solution is continuous with respect to v0 inL²(Rⁿ) for allt≥0.

Let

u(t, ω, u0) =e^εz(θtω)v(t, ω, v0), (3.14) where

v0=e^−εz(ω)u0. (3.15) We can associate a random dynamical system Φεwith problem (3.1)-(3.2) viaufor eachε >0, where Φε:R⁺×Ω×L²(Rⁿ)7→L²(Rⁿ) is given by

Φε(t, ω)u0=u(t, ω, u0), for every (t, ω, u0)∈R⁺×Ω×L²(Rⁿ). (3.16) Then Φεis a continuous random dynamical system over (Ω,F,P,(θt)_t∈R) inL²(Rⁿ).

In the sequel, we always assume that D is the collection of all tempered random subsets ofL²(Rⁿ).

In the following, we will first prove that Φε has a unique D-pullback random attractor {Aε(ω)}ω∈Ω. Whenε = 0, problem (3.1)-(3.2) defines a continuous de- terministic dynamical system Φ inL²(Rⁿ). We useA0to denote the global attractor for the deterministic dynamical system. At last, we will establish the relationship of{Aε(ω)}ω∈ΩandA0 whenε→0.

4. Uniform estimates of solutions

In this section, we derive uniform estimates on the solution of the stochastic p-Laplacian equation onRⁿ whent→ ∞with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness of the random dynamical system associated with the equation. In particular, we will show that the tails of the solutions, i.e.,solutions evaluated at large values of|x|, are uniformly small when time is sufficiently large.

Lemma 4.1. Let 0 < ε ≤ 1, g ∈ L²(Rⁿ) and (3.3)-(3.4) hold. Then for every B = {B(ω)}ω∈Ω ∈ D and P-a.e.ω ∈ Ω, there is T(B, ω) > 0, independent of ε, such that for all u₀(θ_−tω)∈B(θ_−tω)andt≥T(B, ω),

ku(t, θ_−tω, u0(θ_−tω))k²≤ρ²₁(ω) := 1 +ce^2εz(ω)Mε(ω). (4.1) Furthermoreρ1(ω) is a tempered function.

Proof. Multiplying (3.12) byv and then integrating overRⁿ, we find that 1

2 d

dtkvk²=−e^{ε(p−2)z(θtω)} Z

Rⁿ

|∇v|^pdx−λkvk²+e^{−εz(θtω)} Z

Rⁿ

f(x, u)vdx +e^{−εz(θtω)}

Z

Rⁿ

g(x)vdx+εz(θtω)kvk²

. (4.2)

For the nonlinear term, by (3.3), we have e^{−εz(θtω)}

Z

Rⁿ

f(x, u)vdx=e^{−2εz(θtω)} Z

Rⁿ

f(x, u)udx

≤ −α1e^{−2εz(θtω)} Z

Rⁿ

|u|^pdx+e^{−2εz(θtω)} Z

Rⁿ

|ψ1(x)|dx

=−α1e^{−2εz(θtω)}kuk^p_p+e^{−2εz(θtω)}kψ1kL₁,

(4.3) And

e^{−εz(θtω)} Z

Rⁿ

g(x)vdx≤e^{−εz(θtω)}kgk · kvk ≤ 1

2λe^{−2εz(θtω)}kgk²+λ

2kvk². (4.4) Then it from (4.2)-(4.4) it follows that

d

dtkvk²≤ −2e^{ε(p−2)z(θtω)} Z

Rⁿ

|∇v|^pdx+ (2εz(θ_tω)−λ)kvk²

−2α1e^{−2εz(θtω)}kuk^p_p+ (1

λkgk²+ 2kψ1kL₁)e^{−2εz(θtω)}.

(4.5)

Thus,

d

dtkvk²≤(2εz(θtω)−λ)kvk²+ce^{−2εz(θtω)}. (4.6) By the Gronwall Lemma,

kv(t, ω, v0(ω)k²

≤ kv0(ω)k²e^{2εR0tz(θsω)ds−λt}+c Z t

0

e^{−2εz(θsω)+2εRstz(θτ}ω)dτ−λ(t−s)ds.

(4.7)

Replaceω byθ−tω in (4.7), we have

kv(t, θ−tω, v₀(θ_−tω)k²≤ kv0(θ_−tω)k²e^{2εR0tz(θs−tω)ds−λt} +c

Z t

0

e^{−2εz(θs−tω)+2εRstz(θτ−t}ω)dτ−λ(t−s)ds

=kv0(θ−tω)k²e^2ε

R0

−tz(θsω)ds−λt

+c Z 0

−t

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds

≤ kv₀(θ_−tω)k²e^{2εR−t0 z(θsω)ds−λt}+cM_ε(ω),

(4.8)

whereMε(ω) is defined in (3.9).

It follows from (4.8) and (3.11) that ku(t, θ_−tω, u₀(θ_−tω)k²

=ke^εz(ω)v(t, θ_−tω, v₀(θ_−tω)k²

=e^2εz(ω)kv(t, θ−tω, v₀(θ_−tω)k²

≤e^2εz(ω)(kv0(θ_−tω)k²e^{2εR−t0 z(θsω)ds−λt}+cMε(ω))

=e^2εz(ω)(e^{−2εz(θ−tω)}ku0(θ_−tω)k²e^{2εR−t0 z(θsω)ds−λt}+cM_ε(ω)).

(4.9)

SinceB={B(ω)}_ω∈Ω∈ Dandu₀(θ_−tω)∈B(θ_−tω), due to (3.7)(3.8), there exists T(B, ω)>0, independent ofε, such that for allt≥T(B, ω)

ku0(θ_−tω)k²e2εz(ω)−2εz(θ−tω)+2εR0

−tz(θ_sω)ds−λt

≤ ku₀(θ_−tω)k²e2|z(ω)|+2|z(θ−tω)|+2|R0

−tz(θsω)ds|−λt

≤ ku0(θ−tω)k²e^−λ2t≤1,

(4.10)

which along with (4.9) implies that for allt≥T(B, ω)

ku(t, θ_−tω, u0(θ_−tω))k²≤ρ²₁(ω) := 1 +ce^2εz(ω)Mε(ω). (4.11) It is easy to prove thatρ1(ω) is a tempered function.

Lemma 4.2. Let 0 < ε ≤ 1, g ∈ L²(Rⁿ) and (3.3)-(3.4) hold. Then for every B ={B(ω)}_ω∈Ω ∈ D and P-a.e. ω ∈Ω, there is T(B, ω)>0, independent ofε, such that for all u0(θ_−tω)∈B(θ_−tω)andt≥T(B, ω)

Z t+1

t

k∇u(s, θ−t−1ω, u₀(θ_−t−1ω))k^p_p≤1 +ce^2pεK(ω)M_ε(ω), (4.12) Z t+1

t−1

ku(s, θ−t−1ω, u₀(θ_−t−1ω))k^p_p≤1 +ce^6εK(ω)M_ε(ω). (4.13) Proof. From (4.5), we have

d

dtkvk²≤ −2e^{ε(p−2)z(θtω)}k∇vk^p_pdx+ (2εz(θtω)−λ)kvk²

−2α₁e^{−2εz(θtω)}kuk^p_p+ce^{−2εz(θtω)}.

(4.14)

Using Gronwall Lemma, for allt≥T≥0, we have kv(t, ω, v0(ω))k²

≤e^{2εRTtz(θsω)ds−λ(t−T)}kv(T, ω, v0(ω))k² +c

Z t

T

e^{−2εz(θsω)+2εRstz(θτ}ω)dτ+λ(s−t)

−2α1

Z t

T

e^{−2εz(θsω)+2εRstz(θτ}ω)dτ+λ(s−t)ku(s, ω, u0(ω))k^p_pds

−2 Z t

T

e^{ε(p−2)z(θsω)+2εRstz(θτ}ω)dτ+λ(s−t)k∇v(s, ω, v0(ω))k^p_pds.

(4.15)

Replaceω byθ_−tω andtbyT in (4.7), we have

kv(T, θ_−tω, v0(θ_−tω)k²≤ kv0(θ_−tω)k²e^{2εR0Tz(θs−tω)ds−λT} +c

Z T

0

e^{−2εz(θs−tω)+2εRsTz(θτ−tω)dτ−λ(T−s)}ds.

(4.16)

Multiplying the two sides of (4.16) bye^{2εRTtz(θs−tω)ds−λ(t−T)}, then for allt≥T, e^{2εRTtz(θs−tω)ds−λ(t−T)}kv(T, θ−tω, v₀(θ_−tω)k²

≤ kv₀(θ_−tω)k²e^{2εR0tz(θs−tω)ds−λt}+c Z T

0

e^{−2εz(θs−tω)+2εRstz(θτ−t}ω)dτ−λ(t−s)ds.

(4.17) Thus, replaceω byθ−tω in (4.15) and together with (4.17), it follows that

2α₁ Z t

T

e^{−2εz(θs−tω)+2εRstz(θτ−t}ω)dτ+λ(s−t)ku(s, θ_−tω, u₀(θ_−tω))k^p_pds + 2

Z t

T

e^{ε(p−2)z(θs−tω)+2εRstz(θτ−t}ω)dτ+λ(s−t)k∇v(s, θ_−tω, v0(θ_−tω))k^p_pds

≤ kv0(θ−tω)k²e^{2εR0tz(θs−tω)ds−λt}+c Z t

0

e^{−2εz(θs−tω)+2εRstz(θτ−t}ω)dτ−λ(t−s)ds

=kv0(θ−tω)k²e^2ε

R0

−tz(θsω)ds−λt

+c Z 0

−t

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds.

(4.18) Replacetbyt+ 1 andT bytin (4.18), we have

2 Z t+1

t

e^{ε(p−2)z(θs−t−1ω)+2εRst+1z(θτ−t−1}ω)dτ+λ(s−t−1)

× k∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds

≤ kv0(θ_−t−1ω)k²e^{2εR−t−10 z(θs}ω)ds−λ(t+1)+c Z 0

−t−1

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds.

Using (3.10), we have Z t+1

t

e^{ε(p−2)z(θs−t−1ω)+2εRst+1z(θτ−t−1}ω)dτ+λ(s−t−1)

× k∇v(s, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pds

≥ Z t+1

t

e−ε(p−2)K(ω)−2εK(ω)−λk∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds

=e^{−pεK(ω)−λ} Z t+1

t

k∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds.

Thus Z t+1

t

k∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds

≤ 1

2kv0(θ_−t−1ω)k²e^{pεK(ω)+2εR−t−10 z(θsω)ds−λt} +ce^pεK(ω)+λ

Z 0

−t−1

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds

≤ 1

2kv0(θ−t−1ω)k²e^pεK(ω)+2ε

R0

−t−1z(θsω)ds−λt

+ce^pεK(ω)+λMε(ω).

(4.19)

It follows from (4.19) that Z t+1

t

k∇u(s, θ_−t−1ω, u0(θ_−t−1ω))k^p_pds

= Z t+1

t

e^{pεz(θs−t−1ω)}k∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds

≤e^pεK(ω) Z t+1

t

k∇v(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds

≤ 1

2kv0(θ_−t−1ω)k²e^{2pεK(ω)+2εR−t−10 z(θsω)ds−λt}+ce^2pεK(ω)+λMε(ω)

≤cku0(θ−t−1ω)k²e^{−2εz(θ−t−1}ω)+2pεK(ω)+2εR0

−t−1z(θ_sω)ds−λt

+ce^2pεK(ω)+λMε(ω).

(4.20)

On the other hand, since Z t+1

t−1

e^{−2εz(θs−t−1ω)+2εRst+1z(θτ−t−1}ω)dτ+λ(s−t−1)ku(s, θ_−t−1ω, u₀(θ_−t−1ω))k^p_pds

≥e^{−6εK(ω)−2λ} Z t+1

t−1

ku(s, θ−t−1ω, u₀(θ_−t−1ω))k^p_pds.

(4.21) Replacetbyt+ 1 andT byt−1 in (4.18) and using (4.21), we obtain

Z t+1

t−1

ku(s, θ_−t−1ω, u0(θ_−t−1ω))k^p_pds

≤ckv0(θ_−t−1ω)k²e^{6εK(ω)+2εR−t−10 z(θsω)ds−λt+λ} +ce^6εK(ω)+2λ

Z 0

−t−1

e^{−2εz(θsω)+2εRs0z(θτω)dτ+λs}ds

≤ckv0(θ−t−1ω)k²e^6εK(ω)+2ε

R0

−t−1z(θ_sω)ds−λt+λ

+ce^6εK(ω)+2λMε(ω)

≤cku₀(θ_−t−1ω)k²e^{−2εz(θ−t−1}ω)+6εK(ω)+2εR0

−t−1z(θ_sω)ds−λt+λ

+ce^6εK(ω)+2λM_ε(ω).

(4.22)

Since B ={B(ω)}ω∈Ω ∈ D and u0(θ−t(ω))∈ B(θ−t(ω)), similar to (4.10), there existsT(B, ω)>0, independent ofε, such that for allt≥T(B, ω)

cku0(θ_−t−1ω)k²e^{−2εz(θ−t−1}ω)+2pεK(ω)+2εR0

−t−1z(θ_sω)ds−λt

≤1, (4.23)

cku₀(θ_−t−1ω)k²e^{−2εz(θ−t−1}ω)+6εK(ω)+2εR0

−t−1z(θ_sω)ds−λt+λ≤1. (4.24) From (4.20), (4.22) and using (4.23), (4.24), we obtain that for allt≥T(B, ω),

Z t+1

t

k∇u(s, θ_−t−1ω, u0(θ_−t−1ω))k^p_p≤1 +ce^2pεK(ω)Mε(ω), (4.25) Z t+1

t−1

ku(s, θ_−t−1ω, u0(θ_−t−1ω))k^p_p≤1 +ce^6εK(ω)Mε(ω). (4.26) Lemma 4.3. Let 0 < ε ≤ 1, g ∈ L²(Rⁿ) and (3.3)-(3.4) hold. Then for every B ={B(ω)}_ω∈Ω ∈ D and P-a.e. ω ∈Ω, there is T(B, ω)>0, independent ofε, such that for all u0(θ−tω)∈B(θ−tω)andt≥T(B, ω),∀τ ∈[t, t+ 1]

kv(τ, θ−t−1ω, v0(θ−t−1ω))k^p_p≤ce^pεK(ω)(εK(ω) + 1)(e^6εK(ω)Mε(ω) + 1). (4.27) Proof. Multiplying (3.12) with|v|^p−2v and then integrating overRⁿ, it yields that

1 p

d

dtkvk^p_p=e^{ε(p−2)z(θtω)}(div(|∇v|^p−2∇v),|v|^p−2v)−λkvk^p_p +e^{−εz(θtω)}(f(x, u),|v|^p−2v) +e^{−εz(θtω)}(g(x),|v|^p−2v) +εz(θtω)kvk^p_p.

(4.28)

We now estimate every term of (4.28). First by our assumptionp≥2, we have e^{ε(p−2)z(θtω)}(div(|∇v|^p−2∇v),|v|^p−2v)

=e^{ε(p−2)z(θtω)} Z

Rⁿ n

X

i=1

∂

∂x_i(|∇v|^p−2 ∂v

∂x_i)|v|^p−2vdx

=−e^{ε(p−2)z(θtω)}

n

X

i=1

[ Z

Rⁿ

(|∇v|^p−2 ∂v

∂x_i)(p−2)|v|^p−2∂v

∂x_idx +

Z

Rⁿ

(|∇v|^p−2 ∂v

∂xi

)|v|^p−2 ∂v

∂xi

dx]

=−e^{ε(p−2)z(θtω)}(p−1) Z

Rⁿ

|∇v|^p|v|^p−2dx≤0.

(4.29)

To estimate the nonlinear term, from (3.3), we have

f(x, u)v=e^{−εz(θtω)}f(x, u)u≤ −α₁e^{−εz(θtω)}|u|^p+e^{−εz(θtω)}ψ₁(x)

=−α₁e^{(p−1)εz(θtω)}|v|^p+e^{−εz(θtω)}ψ₁(x). (4.30)

From which it follows by Young’s inequality that e^{−εz(θtω)}(f(x, u),|v|^p−2v)

=e^{−εz(θtω)} Z

Rⁿ

f(x, u)|v|^p−2v

≤ −α1e^{(p−2)εz(θtω)}kvk^2p−2_2p−2+e^{−2εz(θtω)} Z

Rⁿ

ψ1(x)|v|^p−2dx

≤ −α1e^{(p−2)εz(θtω)}kvk^2p−2_2p−2+ce^{−pεz(θtω)}kψ1k

p p2 2

+(p−1)λ p kvk^p_p.

(4.31)

On the other hand, the fourth term on the right-hand side of (4.28) is bounded by

|e^{−εz(θtω)}(g(x),|v|^p−2v)| ≤ α₁

2 e^{(p−2)εz(θtω)}kvk^2p−2_2p−2+ce^{−pεz(θtω)}kgk². (4.32) Then it follows from (4.28)(4.29), (4.31),(4.32) that

d

dtkvk^p_p≤(pεz(θ_tω)−λ)kvk^p_p+ce^{−pεz(θtω)}. (4.33) Integrating (4.33) froms(t−1≤s≤t) toτ(t≤τ≤t+ 1), we obtain

kv(τ, ω, v0(ω))k^p_p

≤ kv(s, ω, v0(ω))k^p_p+ Z τ

s

|pεz(θs⁰ω)−λ|kv(s⁰, ω, v0(ω))k^p_pds⁰ +c

Z τ

s

e^{−pεz(θs0ω)}ds⁰.

(4.34)

Replaceω byθ−t−1ω in (4.34), we have kv(τ, θ−t−1ω, v0(θ−t−1ω))k^p_p

≤ kv(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_p+c Z t+1

t−1

e^{−pεz(θs0 −t−1ω)}ds⁰

+ Z t+1

t−1

|pεz(θs⁰−t−1ω)−λ|kv(s⁰, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pds⁰.

(4.35)

Integrating (4.35) with respect tosfromt−1 tot, we obtain that for allτ∈[t, t+1], kv(τ, θ_−t−1ω, v0(θ_−t−1ω))k^p_p

≤ Z t

t−1

kv(s, θ−t−1ω, v₀(θ_−t−1ω))k^p_pds+c Z 0

−2

e^{−pεz(θsω)}ds

+c(εK(ω) + 1) Z t+1

t−1

kv(s⁰, θ−t−1ω, v0(θ−t−1ω))k^p_pds⁰

≤ Z t+1

t−1

kv(s, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds+ce^pεK(ω) (4.36) +c(εK(ω) + 1)

Z t+1

t−1

kv(s⁰, θ_−t−1ω, v0(θ_−t−1ω))k^p_pds⁰

≤ Z t+1

t−1

e^{−pεz(θs−t−1ω)}ku(s, θ_−t−1ω, u₀(θ_−t−1ω))k^p_pds+ce^pεK(ω) +c(εK(ω) + 1)

Z t+1

t−1

e^{−pεz(θs0 −t−1ω)}ku(s⁰, θ−t−1ω, u0(θ−t−1ω))k^p_pds⁰

≤c(εK(ω) + 1)e^pεK(ω) Z t+1

t−1

ku(s⁰, θ_−t−1ω, u0(θ_−t−1ω))k^p_pds⁰+ce^pεK(ω). (4.37) LetT(B, ω) be the positive constant in Lemma 4.2 andt≥T(B, ω), together with (4.13) and (4.37), we have fort≤τ ≤t+ 1

kv(τ, θ_−t−1ω, v0(θ_−t−1ω))k^p_p≤ce^pεK(ω)(εK(ω) + 1)(e^6εK(ω)Mε(ω) + 1). (4.38) Lemma 4.4. Let 0 < ε ≤ 1, g ∈ L²(Rⁿ) and (3.3)-(3.4) hold. Then for every B = {B(ω)}_ω∈Ω ∈ D and P-a.e.ω ∈ Ω, there is T(B, ω) > 0, independent of ε, such that for all u0(θ_−tω)∈B(θ_−tω)andt≥T(B, ω)

Z t+1

t

ku(s, θ−t−1ω, u0(θ−t−1ω))k^2p−2_2p−2ds≤ce^cεK(ω)(εK(ω) + 1)(e^cεK(ω)Mε(ω) + 1).

(4.39) Proof. Using (4.28) together with (4.29) (4.31)and (4.32), we have

d

dtkvk^p_p≤ −ce^{ε(p−2)z(θtω)}kvk^2p−2_2p−2+ (pεz(θtω)−λ)kvk^p_p+ce^{−pεz(θtω)}. (4.40) Using Gronwall’s Lemma, for allt≥T ≥0

kv(t, ω, v0(ω))k^p_p

≤ kv(T, ω, v0(ω))k^p_pe^{pεRTtz(θsω)ds−λ(t−T)} +c

Z t

T

e^{−pεz(θsω)+pεRstz(θτ}ω)dτ−λ(t−s)ds

−c Z t

T

e^{ε(p−2)z(θsω)+pεRstz(θτ}ω)dτ−λ(t−s)kv(s, ω, v₀(ω))k^2p−2_2p−2ds.

(4.41)

Replaceω byθ−tω in (4.41). It follows that c

Z t

T

e^{ε(p−2)z(θs−tω)+pεRstz(θτ−t}ω)dτ−λ(t−s)kv(s, θ_−tω, v0(θ_−tω))k^2p−2_2p−2ds

≤ kv(T, θ_−tω, v0(θ_−tω))k^p_pe^{pεRTtz(θs−tω)ds−λ(t−T)} +c

Z t

T

e^{−pεz(θs−tω)+pεRstz(θτ−t}ω)dτ−λ(t−s)ds.

(4.42)

Replacingt byt+ 1 andT byt, we have c

Z t+1

t

e^{ε(p−2)z(θs−t−1ω)+pεRst+1z(θτ−t−1}ω)dτ−λ(t+1−s)

× kv(s, θ_−t−1ω, v₀(θ_−t−1ω))k^2p−2_2p−2ds

≤ kv(t, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pe^{pεRtt+1z(θs−t−1ω)ds−λ} +c

Z t+1

t

e^{−pεz(θs−t−1ω)+pεRst+1z(θτ−t−1}ω)dτ−λ(t+1−s)ds

≤ kv(t, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pe^{pεK(ω)−λ}+ce^2pεK(ω).

(4.43)

Note that Z t+1

t

e^{ε(p−2)z(θs−t−1ω)+pεRst+1z(θτ−t−1}ω)dτ−λ(t+1−s)

× kv(s, θ−t−1ω, v0(θ−t−1ω))k^2p−2_2p−2ds

≥e−2pεK(ω)+2εK(ω)−λZ t+1 t

kv(s, θ_−t−1ω, v0(θ_−t−1ω))k^2p−2_2p−2ds.

(4.44)

Thus together with (4.43)-(4.44), we obtain Z t+1

t

ku(s, θ_−t−1ω, u0(θ_−t−1ω))k^2p−2_2p−2ds

= Z t+1

t

e^{εz(θs−t−1ω)(2p−2)}kv(s, θ_−t−1ω, v0(θ_−t−1ω))k^2p−2_2p−2ds

≤e(2p−2)εK(ω)Z t+1 t

kv(s, θ_−t−1ω, v0(θ_−t−1ω))k^2p−2_2p−2ds

≤ce5pεK(ω)−4εK(ω)kv(t, θ−t−1ω, v₀(θ_−t−1ω))k^p_p+ce6pεK(ω)−4εK(ω)+λ.

(4.45)

Let T(B, ω) be the positive constant in Lemma 4.3. Then fort ≥T(B, ω), from (4.27), we have

Z t+1

t

ku(s, θ_−t−1ω, u0(θ_−t−1ω))k^2p−2_2p−2ds

≤ce6pεK(ω)−4εK(ω)(εK(ω) + 1)(e^6εK(ω)M_ε(ω) + 1) +ce6pεK(ω)−4εK(ω)+λ

=ce^cεK(ω)(εK(ω) + 1)(e^cεK(ω)Mε(ω) + 1).

(4.46)

Lemma 4.5. Let 0 < ε ≤ 1, g ∈ L²(Rⁿ) and (3.3)-(3.4) hold. Then for every B ={B(ω)}_ω∈Ω ∈ D and P-a.e. ω ∈Ω, there is T(B, ω)>0, independent ofε, such that for all u0(θ−tω)∈B(θ−tω)andt≥T(B, ω)

k∇u(t, θ−tω, u₀(θ_−tω))k^p_p≤ce^cεK(ω)(εK(ω) + 1)(M_ε(ω) + 1). (4.47) Proof. Take the inner product of (3.12) withvtinL²(Rⁿ), we obtain

kv_tk²=−e^{ε(p−2)z(θtω)}1 p

d

dtk∇vk^p_p−λ(v, v_t) +e^{−εz(θtω)}(f(x, u), v_t) +e^{−εz(θtω)}(g(x), v_t) +εz(θ_tω)(v, v_t).

(4.48) By (3.4), the Cauchy-Schwartz inequality and the Young inequality, we find that

|e^{−εz(θtω)}(f(x, u), v_t)|

≤e^{−2εz(θtω)}kf(x, u)k²+1 4kv_tk²

≤2α²₂e^{−2εz(θtω)}kuk^2p−2_2p−2+ 2e^{−2εz(θtω)}kψ₂k²+1 4kv_tk²,

(4.49)

|e^{−εz(θtω)}(g(x), v_t)| ≤e^{−2εz(θtω)}kgk²+1

4kv_tk², (4.50)

|(εz(θ_tω)−λ)(v, v_t)| ≤ |εz(θ_tω)−λ|²kvk²+1

4kv_tk². (4.51)

It follows from (4.48)-(4.51) that d

dtk∇vk^p_p

≤2pα²₂e^{−pεz(θtω)}kuk^2p−2_2p−2+ 2pe^{−pεz(θtω)}kψ2k²

+pe^{−pεz(θtω)}kgk²+pe^{−ε(p−2)z(θtω)}|εz(θtω)−λ|²kvk²

=ce^{−pεz(θtω)}kuk^2p−2_2p−2+ce^{−pεz(θtω)}+ce^{ε(2−p)z(θtω)}|εz(θtω)−λ|²kvk².

(4.52)

Let T0(B, ω) be the positive constant in Lemma 4.2, take t ≥ T0(B, ω) and s ∈ (t, t+ 1). Integrate (4.52) over (s, t+ 1) to get

k∇v(t+ 1, ω, v₀(ω))k^p_p

≤ k∇v(s, ω, v0(ω))k^p_p+c Z t+1

s

e^{−pεz(θτω)}ku(τ, ω, u0(ω))k^2p−2_2p−2dτ +c

Z t+1

s

e^{−pεz(θτω)}dτ+c Z t+1

s

e^{ε(2−p)z(θτω)}|εz(θτω)−λ|²kv(τ, ω, v0(ω))k²dτ.

(4.53) Integrating with respect tosover (t, t+ 1), it follows that

k∇v(t+ 1, ω, v0(ω))k^p_p

≤ Z t+1

t

k∇v(s, ω, v₀(ω))k^p_pds+c Z t+1

t

e^{−pεz(θτω)}ku(τ, ω, u₀(ω))k^2p−2_2p−2dτ +c

Z t+1

t

e^{−pεz(θτω)}dτ+c Z t+1

t

e^{ε(2−p)z(θτω)}|εz(θ_τω)−λ|²kv(τ, ω, v₀(ω))k²dτ.

(4.54) Replaceω byθ_−t−1ω in the above inequality, we have

k∇v(t+ 1, θ_−t−1ω, v0(θ_−t−1ω))k^p_p

≤ Z t+1

t

k∇v(s, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pds +c

Z t+1

t

e^{−pεz(θτ−t−1ω)}ku(τ, θ_−t−1ω, u₀(θ_−t−1ω))k^2p−2_2p−2dτ +c

Z t+1

t

e^{−pεz(θτ−t−1ω)}dτ

+c Z t+1

t

e^{ε(2−p)z(θτ−t−1ω)}|εz(θ_τ−t−1ω)−λ|²kv(τ, θ_−t−1ω, v₀(θ_−t−1ω))k²dτ

≤ Z t+1

t

k∇v(s, θ_−t−1ω, v₀(θ_−t−1ω))k^p_pds +ce^pεK(ω)+ce^pεK(ω)

Z t+1

t

ku(τ, θ_−t−1ω, u₀(θ_−t−1ω))k^2p−2_2p−2dτ +c(K²(ω) + 1)

Z t+1

t

e^{ε(2−p)z(θτ−t−1ω)}kv(τ, θ_−t−1ω, v₀(θ_−t−1ω))k²dτ,

(4.55) where we used 0< ε≤1.