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 onR^{n}. 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 spaceR^{n}:

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 onR^{n},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 R^{n}. 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 onR^{n}, 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
R^{n}. 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 inL^{2}(R^{n}) respectively
andk · k_{p} to denote the norm inL^{p}(R^{n}). 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∈Ω}of X is called tempered with
respect to (θt)_{t∈}_{R}if for P-a.e.ω∈Ω,

t→∞lim e^{−βt}d(B(θ_{−t}ω)) = 0 for allβ >0,
whered(B) = sup_{x∈B}kxk_{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_{−t}_{n}ω, xn)}^{∞}_{n=1}
has a convergent subsequence in X whenever tn → ∞, and xn ∈ B(θ_{−t}ω) 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_{−t}ω, B(θ_{−t}ω)),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≥T}ϕ(t, θ_{−t}ω, K(θ_{−t}ω)).

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 ϕ_{0} is a deterministic
dynamical system defined on X which has a global attractor A_{0}. 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_{σ}_{n}(t, ω, x_{n}) =ϕ_{0}(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∈E}_{σ}_{(ω)}kxkX.

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

∪_{0<σ≤σ}_{0}A_{σ}(ω) is precompact in X. (2.3)
Then for P-a.e. ω∈Ω,

dist(A_{σ}(ω),A_{0})→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 onR^{n} with multiplicative noise:

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

u(x,0) =u0(x), x∈R^{n}. (3.2)

whereε >0,λ >0,p≥2 are constants,g∈L^{2}(R^{n}),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∈R^{n} and
s∈R^{n},

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∈L^{1}(R^{n})∩L^{p}^{2}(R^{n}),ψ2∈L^{2}(R^{n})∩L^{q}(R^{n})
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∈R^{n}. (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(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds

≤ Z 0

−∞

e^{2|z(θ}^{s}^{ω)|+2|}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)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(θ}^{t}^{ω)}u(t), (3.11)
whereuis a solution of problem (3.1)-(3.2). Thenv satisfies

dv

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

v(x,0) =v0(x) =e^{−εz(ω)}u0(x), x∈R^{n}. (3.13)
Using the standard Galerkin method, one may show that for all v0 ∈L^{2}(R^{n}),
problem (3.12)-(3.13) has a unique solution

v(·, ω, v0)∈C([0,∞), L^{2}(R^{n}))∩L^{2}((0, T), W^{1,p}(R^{n})).

Furthermore, the solution is continuous with respect to v0 inL^{2}(R^{n}) for allt≥0.

Let

u(t, ω, u0) =e^{εz(θ}^{t}^{ω)}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×L^{2}(R^{n})7→L^{2}(R^{n}) is given by

Φε(t, ω)u0=u(t, ω, u0), for every (t, ω, u0)∈R^{+}×Ω×L^{2}(R^{n}). (3.16)
Then Φεis a continuous random dynamical system over (Ω,F,P,(θt)_{t∈R}) inL^{2}(R^{n}).

In the sequel, we always assume that D is the collection of all tempered random
subsets ofL^{2}(R^{n}).

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 inL^{2}(R^{n}). 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 onR^{n} 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 ∈ L^{2}(R^{n}) 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 u_{0}(θ_{−t}ω)∈B(θ_{−t}ω)andt≥T(B, ω),

ku(t, θ_{−t}ω, u0(θ_{−t}ω))k^{2}≤ρ^{2}_{1}(ω) := 1 +ce^{2εz(ω)}Mε(ω). (4.1)
Furthermoreρ1(ω) is a tempered function.

Proof. Multiplying (3.12) byv and then integrating overR^{n}, we find that
1

2 d

dtkvk^{2}=−e^{ε(p−2)z(θ}^{t}^{ω)}
Z

R^{n}

|∇v|^{p}dx−λkvk^{2}+e^{−εz(θ}^{t}^{ω)}
Z

R^{n}

f(x, u)vdx
+e^{−εz(θ}^{t}^{ω)}

Z

R^{n}

g(x)vdx+εz(θtω)kvk^{2}

. (4.2)

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

Z

R^{n}

f(x, u)vdx=e^{−2εz(θ}^{t}^{ω)}
Z

R^{n}

f(x, u)udx

≤ −α1e^{−2εz(θ}^{t}^{ω)}
Z

R^{n}

|u|^{p}dx+e^{−2εz(θ}^{t}^{ω)}
Z

R^{n}

|ψ1(x)|dx

=−α1e^{−2εz(θ}^{t}^{ω)}kuk^{p}_{p}+e^{−2εz(θ}^{t}^{ω)}kψ1kL_{1},

(4.3) And

e^{−εz(θ}^{t}^{ω)}
Z

R^{n}

g(x)vdx≤e^{−εz(θ}^{t}^{ω)}kgk · kvk ≤ 1

2λe^{−2εz(θ}^{t}^{ω)}kgk^{2}+λ

2kvk^{2}. (4.4)
Then it from (4.2)-(4.4) it follows that

d

dtkvk^{2}≤ −2e^{ε(p−2)z(θ}^{t}^{ω)}
Z

R^{n}

|∇v|^{p}dx+ (2εz(θ_{t}ω)−λ)kvk^{2}

−2α1e^{−2εz(θ}^{t}^{ω)}kuk^{p}_{p}+ (1

λkgk^{2}+ 2kψ1kL_{1})e^{−2εz(θ}^{t}^{ω)}.

(4.5)

Thus,

d

dtkvk^{2}≤(2εz(θtω)−λ)kvk^{2}+ce^{−2εz(θ}^{t}^{ω)}. (4.6)
By the Gronwall Lemma,

kv(t, ω, v0(ω)k^{2}

≤ kv0(ω)k^{2}e^{2ε}^{R}^{0}^{t}^{z(θ}^{s}^{ω)ds−λt}+c
Z t

0

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ−λ(t−s)ds.

(4.7)

Replaceω byθ−tω in (4.7), we have

kv(t, θ−tω, v_{0}(θ_{−t}ω)k^{2}≤ kv0(θ_{−t}ω)k^{2}e^{2ε}^{R}^{0}^{t}^{z(θ}^{s−t}^{ω)ds−λt}
+c

Z t

0

e^{−2εz(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ−λ(t−s)ds

=kv0(θ−tω)k^{2}e^{2ε}

R0

−tz(θsω)ds−λt

+c Z 0

−t

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds

≤ kv_{0}(θ_{−t}ω)k^{2}e^{2ε}^{R}^{−t}^{0} ^{z(θ}^{s}^{ω)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_{−t}ω, u_{0}(θ_{−t}ω)k^{2}

=ke^{εz(ω)}v(t, θ_{−t}ω, v_{0}(θ_{−t}ω)k^{2}

=e^{2εz(ω)}kv(t, θ−tω, v_{0}(θ_{−t}ω)k^{2}

≤e^{2εz(ω)}(kv0(θ_{−t}ω)k^{2}e^{2ε}^{R}^{−t}^{0} ^{z(θ}^{s}^{ω)ds−λt}+cMε(ω))

=e^{2εz(ω)}(e^{−2εz(θ}^{−t}^{ω)}ku0(θ_{−t}ω)k^{2}e^{2ε}^{R}^{−t}^{0} ^{z(θ}^{s}^{ω)ds−λt}+cM_{ε}(ω)).

(4.9)

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

ku0(θ_{−t}ω)k^{2}e2εz(ω)−2εz(θ−tω)+2εR0

−tz(θ_{s}ω)ds−λt

≤ ku_{0}(θ_{−t}ω)k^{2}e2|z(ω)|+2|z(θ−tω)|+2|R0

−tz(θsω)ds|−λt

≤ ku0(θ−tω)k^{2}e^{−}^{λ}^{2}^{t}≤1,

(4.10)

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

ku(t, θ_{−t}ω, u0(θ_{−t}ω))k^{2}≤ρ^{2}_{1}(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 ∈ L^{2}(R^{n}) 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_{−t}ω)∈B(θ_{−t}ω)andt≥T(B, ω)

Z t+1

t

k∇u(s, θ−t−1ω, u_{0}(θ_{−t−1}ω))k^{p}_{p}≤1 +ce^{2pεK(ω)}M_{ε}(ω), (4.12)
Z t+1

t−1

ku(s, θ−t−1ω, u_{0}(θ_{−t−1}ω))k^{p}_{p}≤1 +ce^{6εK(ω)}M_{ε}(ω). (4.13)
Proof. From (4.5), we have

d

dtkvk^{2}≤ −2e^{ε(p−2)z(θ}^{t}^{ω)}k∇vk^{p}_{p}dx+ (2εz(θtω)−λ)kvk^{2}

−2α_{1}e^{−2εz(θ}^{t}^{ω)}kuk^{p}_{p}+ce^{−2εz(θ}^{t}^{ω)}.

(4.14)

Using Gronwall Lemma, for allt≥T≥0, we have
kv(t, ω, v0(ω))k^{2}

≤e^{2ε}^{R}^{T}^{t}^{z(θ}^{s}^{ω)ds−λ(t−T}^{)}kv(T, ω, v0(ω))k^{2}
+c

Z t

T

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ+λ(s−t)

−2α1

Z t

T

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ+λ(s−t)ku(s, ω, u0(ω))k^{p}_{p}ds

−2 Z t

T

e^{ε(p−2)z(θ}^{s}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ+λ(s−t)k∇v(s, ω, v0(ω))k^{p}_{p}ds.

(4.15)

Replaceω byθ_{−t}ω andtbyT in (4.7), we have

kv(T, θ_{−t}ω, v0(θ_{−t}ω)k^{2}≤ kv0(θ_{−t}ω)k^{2}e^{2ε}^{R}^{0}^{T}^{z(θ}^{s−t}^{ω)ds−λT}
+c

Z T

0

e^{−2εz(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{T}^{z(θ}^{τ−t}^{ω)dτ−λ(T}^{−s)}ds.

(4.16)

Multiplying the two sides of (4.16) bye^{2ε}^{R}^{T}^{t}^{z(θ}^{s−t}^{ω)ds−λ(t−T}^{)}, then for allt≥T,
e^{2ε}^{R}^{T}^{t}^{z(θ}^{s−t}^{ω)ds−λ(t−T}^{)}kv(T, θ−tω, v_{0}(θ_{−t}ω)k^{2}

≤ kv_{0}(θ_{−t}ω)k^{2}e^{2ε}^{R}^{0}^{t}^{z(θ}^{s−t}^{ω)ds−λt}+c
Z T

0

e^{−2εz(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ−λ(t−s)ds.

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

2α_{1}
Z t

T

e^{−2εz(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ+λ(s−t)ku(s, θ_{−t}ω, u_{0}(θ_{−t}ω))k^{p}_{p}ds
+ 2

Z t

T

e^{ε(p−2)z(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ+λ(s−t)k∇v(s, θ_{−t}ω, v0(θ_{−t}ω))k^{p}_{p}ds

≤ kv0(θ−tω)k^{2}e^{2ε}^{R}^{0}^{t}^{z(θ}^{s−t}^{ω)ds−λt}+c
Z t

0

e^{−2εz(θ}^{s−t}^{ω)+2ε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ−λ(t−s)ds

=kv0(θ−tω)k^{2}e^{2ε}

R0

−tz(θsω)ds−λt

+c Z 0

−t

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds.

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

2 Z t+1

t

e^{ε(p−2)z(θ}^{s−t−1}^{ω)+2ε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ+λ(s−t−1)

× k∇v(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds

≤ kv0(θ_{−t−1}ω)k^{2}e^{2ε}^{R}^{−t−1}^{0} ^{z(θ}^{s}ω)ds−λ(t+1)+c
Z 0

−t−1

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds.

Using (3.10), we have Z t+1

t

e^{ε(p−2)z(θ}^{s−t−1}^{ω)+2ε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ+λ(s−t−1)

× k∇v(s, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}ds

≥ Z t+1

t

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

=e^{−pεK(ω)−λ}
Z t+1

t

k∇v(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds.

Thus Z t+1

t

k∇v(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds

≤ 1

2kv0(θ_{−t−1}ω)k^{2}e^{pεK(ω)+2ε}^{R}^{−t−1}^{0} ^{z(θ}^{s}^{ω)ds−λt}
+ce^{pεK(ω)+λ}

Z 0

−t−1

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds

≤ 1

2kv0(θ−t−1ω)k^{2}e^{pεK(ω)+2ε}

R0

−t−1z(θsω)ds−λt

+ce^{pεK(ω)+λ}Mε(ω).

(4.19)

It follows from (4.19) that Z t+1

t

k∇u(s, θ_{−t−1}ω, u0(θ_{−t−1}ω))k^{p}_{p}ds

= Z t+1

t

e^{pεz(θ}^{s−t−1}^{ω)}k∇v(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds

≤e^{pεK(ω)}
Z t+1

t

k∇v(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds

≤ 1

2kv0(θ_{−t−1}ω)k^{2}e^{2pεK(ω)+2ε}^{R}^{−t−1}^{0} ^{z(θ}^{s}^{ω)ds−λt}+ce^{2pεK(ω)+λ}Mε(ω)

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

−t−1z(θ_{s}ω)ds−λt

+ce^{2pεK(ω)+λ}Mε(ω).

(4.20)

On the other hand, since Z t+1

t−1

e^{−2εz(θ}^{s−t−1}^{ω)+2ε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ+λ(s−t−1)ku(s, θ_{−t−1}ω, u_{0}(θ_{−t−1}ω))k^{p}_{p}ds

≥e^{−6εK(ω)−2λ}
Z t+1

t−1

ku(s, θ−t−1ω, u_{0}(θ_{−t−1}ω))k^{p}_{p}ds.

(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−1}ω, u0(θ_{−t−1}ω))k^{p}_{p}ds

≤ckv0(θ_{−t−1}ω)k^{2}e^{6εK(ω)+2ε}^{R}^{−t−1}^{0} ^{z(θ}^{s}^{ω)ds−λt+λ}
+ce^{6εK(ω)+2λ}

Z 0

−t−1

e^{−2εz(θ}^{s}^{ω)+2ε}^{R}^{s}^{0}^{z(θ}^{τ}^{ω)dτ+λs}ds

≤ckv0(θ−t−1ω)k^{2}e^{6εK(ω)+2ε}

R0

−t−1z(θ_{s}ω)ds−λt+λ

+ce^{6εK(ω)+2λ}Mε(ω)

≤cku_{0}(θ_{−t−1}ω)k^{2}e^{−2εz(θ}^{−t−1}ω)+6εK(ω)+2εR0

−t−1z(θ_{s}ω)ds−λt+λ

+ce^{6εK(ω)+2λ}M_{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−1}ω)k^{2}e^{−2εz(θ}^{−t−1}ω)+2pεK(ω)+2εR0

−t−1z(θ_{s}ω)ds−λt

≤1, (4.23)

cku_{0}(θ_{−t−1}ω)k^{2}e^{−2εz(θ}^{−t−1}ω)+6εK(ω)+2εR0

−t−1z(θ_{s}ω)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−1}ω, u0(θ_{−t−1}ω))k^{p}_{p}≤1 +ce^{2pεK(ω)}Mε(ω), (4.25)
Z t+1

t−1

ku(s, θ_{−t−1}ω, u0(θ_{−t−1}ω))k^{p}_{p}≤1 +ce^{6εK(ω)}Mε(ω). (4.26)
Lemma 4.3. Let 0 < ε ≤ 1, g ∈ L^{2}(R^{n}) 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−2}v and then integrating overR^{n}, it yields that

1 p

d

dtkvk^{p}_{p}=e^{ε(p−2)z(θ}^{t}^{ω)}(div(|∇v|^{p−2}∇v),|v|^{p−2}v)−λkvk^{p}_{p}
+e^{−εz(θ}^{t}^{ω)}(f(x, u),|v|^{p−2}v) +e^{−εz(θ}^{t}^{ω)}(g(x),|v|^{p−2}v)
+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(θ}^{t}^{ω)}(div(|∇v|^{p−2}∇v),|v|^{p−2}v)

=e^{ε(p−2)z(θ}^{t}^{ω)}
Z

R^{n}
n

X

i=1

∂

∂x_{i}(|∇v|^{p−2} ∂v

∂x_{i})|v|^{p−2}vdx

=−e^{ε(p−2)z(θ}^{t}^{ω)}

n

X

i=1

[ Z

R^{n}

(|∇v|^{p−2} ∂v

∂x_{i})(p−2)|v|^{p−2}∂v

∂x_{i}dx
+

Z

R^{n}

(|∇v|^{p−2} ∂v

∂xi

)|v|^{p−2} ∂v

∂xi

dx]

=−e^{ε(p−2)z(θ}^{t}^{ω)}(p−1)
Z

R^{n}

|∇v|^{p}|v|^{p−2}dx≤0.

(4.29)

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

f(x, u)v=e^{−εz(θ}^{t}^{ω)}f(x, u)u≤ −α_{1}e^{−εz(θ}^{t}^{ω)}|u|^{p}+e^{−εz(θ}^{t}^{ω)}ψ_{1}(x)

=−α_{1}e^{(p−1)εz(θ}^{t}^{ω)}|v|^{p}+e^{−εz(θ}^{t}^{ω)}ψ_{1}(x). (4.30)

From which it follows by Young’s inequality that
e^{−εz(θ}^{t}^{ω)}(f(x, u),|v|^{p−2}v)

=e^{−εz(θ}^{t}^{ω)}
Z

R^{n}

f(x, u)|v|^{p−2}v

≤ −α1e^{(p−2)εz(θ}^{t}^{ω)}kvk^{2p−2}_{2p−2}+e^{−2εz(θ}^{t}^{ω)}
Z

R^{n}

ψ1(x)|v|^{p−2}dx

≤ −α1e^{(p−2)εz(θ}^{t}^{ω)}kvk^{2p−2}_{2p−2}+ce^{−pεz(θ}^{t}^{ω)}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(θ}^{t}^{ω)}(g(x),|v|^{p−2}v)| ≤ α_{1}

2 e^{(p−2)εz(θ}^{t}^{ω)}kvk^{2p−2}_{2p−2}+ce^{−pεz(θ}^{t}^{ω)}kgk^{2}. (4.32)
Then it follows from (4.28)(4.29), (4.31),(4.32) that

d

dtkvk^{p}_{p}≤(pεz(θ_{t}ω)−λ)kvk^{p}_{p}+ce^{−pεz(θ}^{t}^{ω)}. (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(θs^{0}ω)−λ|kv(s^{0}, ω, v0(ω))k^{p}_{p}ds^{0}
+c

Z τ

s

e^{−pεz(θ}^{s}^{0}^{ω)}ds^{0}.

(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−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}+c
Z t+1

t−1

e^{−pεz(θ}^{s0 −t−1}^{ω)}ds^{0}

+ Z t+1

t−1

|pεz(θs^{0}−t−1ω)−λ|kv(s^{0}, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}ds^{0}.

(4.35)

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

≤ Z t

t−1

kv(s, θ−t−1ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}ds+c
Z 0

−2

e^{−pεz(θ}^{s}^{ω)}ds

+c(εK(ω) + 1) Z t+1

t−1

kv(s^{0}, θ−t−1ω, v0(θ−t−1ω))k^{p}_{p}ds^{0}

≤ Z t+1

t−1

kv(s, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds+ce^{pεK(ω)} (4.36)
+c(εK(ω) + 1)

Z t+1

t−1

kv(s^{0}, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}ds^{0}

≤ Z t+1

t−1

e^{−pεz(θ}^{s−t−1}^{ω)}ku(s, θ_{−t−1}ω, u_{0}(θ_{−t−1}ω))k^{p}_{p}ds+ce^{pεK(ω)}
+c(εK(ω) + 1)

Z t+1

t−1

e^{−pεz(θ}^{s0 −t−1}^{ω)}ku(s^{0}, θ−t−1ω, u0(θ−t−1ω))k^{p}_{p}ds^{0}

≤c(εK(ω) + 1)e^{pεK(ω)}
Z t+1

t−1

ku(s^{0}, θ_{−t−1}ω, u0(θ_{−t−1}ω))k^{p}_{p}ds^{0}+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−1}ω, v0(θ_{−t−1}ω))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 ∈ L^{2}(R^{n}) 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_{−t}ω)∈B(θ_{−t}ω)andt≥T(B, ω)

Z t+1

t

ku(s, θ−t−1ω, u0(θ−t−1ω))k^{2p−2}_{2p−2}ds≤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(θ}^{t}^{ω)}kvk^{2p−2}_{2p−2}+ (pεz(θtω)−λ)kvk^{p}_{p}+ce^{−pεz(θ}^{t}^{ω)}. (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}_{p}e^{pε}^{R}^{T}^{t}^{z(θ}^{s}^{ω)ds−λ(t−T}^{)}
+c

Z t

T

e^{−pεz(θ}^{s}^{ω)+pε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ−λ(t−s)ds

−c Z t

T

e^{ε(p−2)z(θ}^{s}^{ω)+pε}^{R}^{s}^{t}^{z(θ}^{τ}ω)dτ−λ(t−s)kv(s, ω, v_{0}(ω))k^{2p−2}_{2p−2}ds.

(4.41)

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

Z t

T

e^{ε(p−2)z(θ}^{s−t}^{ω)+pε}^{R}^{s}^{t}^{z(θ}^{τ−t}ω)dτ−λ(t−s)kv(s, θ_{−t}ω, v0(θ_{−t}ω))k^{2p−2}_{2p−2}ds

≤ kv(T, θ_{−t}ω, v0(θ_{−t}ω))k^{p}_{p}e^{pε}^{R}^{T}^{t}^{z(θ}^{s−t}^{ω)ds−λ(t−T}^{)}
+c

Z t

T

e^{−pεz(θ}^{s−t}^{ω)+pε}^{R}^{s}^{t}^{z(θ}^{τ−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(θ}^{s−t−1}^{ω)+pε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ−λ(t+1−s)

× kv(s, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{2p−2}_{2p−2}ds

≤ kv(t, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}e^{pε}^{R}^{t}^{t+1}^{z(θ}^{s−t−1}^{ω)ds−λ}
+c

Z t+1

t

e^{−pεz(θ}^{s−t−1}^{ω)+pε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ−λ(t+1−s)ds

≤ kv(t, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}e^{pεK(ω)−λ}+ce^{2pεK(ω)}.

(4.43)

Note that Z t+1

t

e^{ε(p−2)z(θ}^{s−t−1}^{ω)+pε}^{R}^{s}^{t+1}^{z(θ}^{τ−t−1}ω)dτ−λ(t+1−s)

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

≥e−2pεK(ω)+2εK(ω)−λZ t+1 t

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

(4.44)

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

t

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

= Z t+1

t

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

≤e(2p−2)εK(ω)Z t+1 t

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

≤ce5pεK(ω)−4εK(ω)kv(t, θ−t−1ω, v_{0}(θ_{−t−1}ω))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−1}ω, u0(θ_{−t−1}ω))k^{2p−2}_{2p−2}ds

≤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 ∈ L^{2}(R^{n}) 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, u_{0}(θ_{−t}ω))k^{p}_{p}≤ce^{cεK(ω)}(εK(ω) + 1)(M_{ε}(ω) + 1). (4.47)
Proof. Take the inner product of (3.12) withvtinL^{2}(R^{n}), we obtain

kv_{t}k^{2}=−e^{ε(p−2)z(θ}^{t}^{ω)}1
p

d

dtk∇vk^{p}_{p}−λ(v, v_{t}) +e^{−εz(θ}^{t}^{ω)}(f(x, u), v_{t})
+e^{−εz(θ}^{t}^{ω)}(g(x), v_{t}) +εz(θ_{t}ω)(v, v_{t}).

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

|e^{−εz(θ}^{t}^{ω)}(f(x, u), v_{t})|

≤e^{−2εz(θ}^{t}^{ω)}kf(x, u)k^{2}+1
4kv_{t}k^{2}

≤2α^{2}_{2}e^{−2εz(θ}^{t}^{ω)}kuk^{2p−2}_{2p−2}+ 2e^{−2εz(θ}^{t}^{ω)}kψ_{2}k^{2}+1
4kv_{t}k^{2},

(4.49)

|e^{−εz(θ}^{t}^{ω)}(g(x), v_{t})| ≤e^{−2εz(θ}^{t}^{ω)}kgk^{2}+1

4kv_{t}k^{2}, (4.50)

|(εz(θ_{t}ω)−λ)(v, v_{t})| ≤ |εz(θ_{t}ω)−λ|^{2}kvk^{2}+1

4kv_{t}k^{2}. (4.51)

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

dtk∇vk^{p}_{p}

≤2pα^{2}_{2}e^{−pεz(θ}^{t}^{ω)}kuk^{2p−2}_{2p−2}+ 2pe^{−pεz(θ}^{t}^{ω)}kψ2k^{2}

+pe^{−pεz(θ}^{t}^{ω)}kgk^{2}+pe^{−ε(p−2)z(θ}^{t}^{ω)}|εz(θtω)−λ|^{2}kvk^{2}

=ce^{−pεz(θ}^{t}^{ω)}kuk^{2p−2}_{2p−2}+ce^{−pεz(θ}^{t}^{ω)}+ce^{ε(2−p)z(θ}^{t}^{ω)}|εz(θtω)−λ|^{2}kvk^{2}.

(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, v_{0}(ω))k^{p}_{p}

≤ k∇v(s, ω, v0(ω))k^{p}_{p}+c
Z t+1

s

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

Z t+1

s

e^{−pεz(θ}^{τ}^{ω)}dτ+c
Z t+1

s

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

(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, v_{0}(ω))k^{p}_{p}ds+c
Z t+1

t

e^{−pεz(θ}^{τ}^{ω)}ku(τ, ω, u_{0}(ω))k^{2p−2}_{2p−2}dτ
+c

Z t+1

t

e^{−pεz(θ}^{τ}^{ω)}dτ+c
Z t+1

t

e^{ε(2−p)z(θ}^{τ}^{ω)}|εz(θ_{τ}ω)−λ|^{2}kv(τ, ω, v_{0}(ω))k^{2}dτ.

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

k∇v(t+ 1, θ_{−t−1}ω, v0(θ_{−t−1}ω))k^{p}_{p}

≤ Z t+1

t

k∇v(s, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}ds
+c

Z t+1

t

e^{−pεz(θ}^{τ−t−1}^{ω)}ku(τ, θ_{−t−1}ω, u_{0}(θ_{−t−1}ω))k^{2p−2}_{2p−2}dτ
+c

Z t+1

t

e^{−pεz(θ}^{τ−t−1}^{ω)}dτ

+c Z t+1

t

e^{ε(2−p)z(θ}^{τ−t−1}^{ω)}|εz(θ_{τ−t−1}ω)−λ|^{2}kv(τ, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{2}dτ

≤ Z t+1

t

k∇v(s, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{p}_{p}ds
+ce^{pεK(ω)}+ce^{pεK(ω)}

Z t+1

t

ku(τ, θ_{−t−1}ω, u_{0}(θ_{−t−1}ω))k^{2p−2}_{2p−2}dτ
+c(K^{2}(ω) + 1)

Z t+1

t

e^{ε(2−p)z(θ}^{τ−t−1}^{ω)}kv(τ, θ_{−t−1}ω, v_{0}(θ_{−t−1}ω))k^{2}dτ,

(4.55) where we used 0< ε≤1.