Dynamics of Non-autonomous Stochastic Semi-linear Degenerate Parabolic Equations with Nonlinear Noise

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Dynamics of Non-autonomous Stochastic Semi-linear Degenerate Parabolic Equations with Nonlinear Noise

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Xin Liu^*,Yan jiao Li

This version is not peer-reviewed

Submitted:

20 June 2023

Posted:

20 June 2023

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Abstract

At present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on bounded or unbounded domain and driven by a nonlinear noise and defined. Since the theory of pathwise random dynamical systems can not be applied directly to the equation with nonlinear noise, first, we establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong-Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong-Zakai approximate system of the equation under consideration driven by a linear multiplicative noise.

Keywords:

Subject: Physical Sciences - Mathematical Physics

MSC: Primary 37L55; 35B40; Secondary 37B55; 35B41

1. Introduction

We consider the following stochastic semi-linear degenerate parabolic equation:

\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - div (σ (x) \nabla u) + λ u + f (x, u) = g (t, x) + h (t, x, u) \frac{d W}{d t}, t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (t, x) |}_{\partial O} = 0, t > τ, \end{matrix} \end{matrix}

(1)

where

O \subseteq R^{N} (N \geq 2)

is an arbitrary (bounded or unbounded) domain,

λ

is positive constants, W is a two-sided Hilbert space valued cylindrical Wiener process or a two-side real-valued Wiener process, the drift term f and diffusion term h are nonlinear functions with respect to u, the given function

g (t, x) \in L_{loc}^{2} (R, L^{2} (O))

. In addition, the variable nonnegative coefficient

σ (x)

is allowed to have at most a finite number of (essential) zeros at some points, which is understood the degeneracy of (1). As in [3,8], we assume that the nonnegative function

σ (x) : O \to R^{+} \cup {0}

satisfies the following hypotheses:

( $H_{α}$ ): $σ \in L_{loc}^{1} (O)$ and for some $α \in (0, 1)$ , $\underset{x \to z}{lim inf} {| x - z |}^{- α} σ (x) > 0$ for every $z \in \bar{O}$ , when the domain $O$ is bounded;
( $H_{α, β}$ ): $σ$ satisfies condition $(H_{α})$ and $\underset{| x | \to \infty}{lim inf} {| x |}^{- β} σ (x) > 0$ for some $β > 2$ , when the domain $O$ is unbounded.

The conditions

(H_{α})

and

(H_{α, β})

indicates that the diffusion coefficient

σ (x)

is extremely irregular.

One of the most important things in studying evolution partial differential equations is to investigate the long-time behavior of solutions of the equations. In this process, attractors are the ideal objects. At present, abundant results, both in abstract context and concrete models, have been established for the deterministic infinite-dimensional dynamical systems, see, e.g. monographs [2,14,25] and papers [3,4,11]. However, when one considers the random influences on the systems under investigation, which are always presented as stochastic partial differential equations, and tries to establish the existence of attractors for them, the theory on deterministic infinite-dimensional system can not be applied directly. On the one hand, the stochastic dynamical systems are non-autonomous, and one can not obtain uniform (w.r.t stochastic time symbol) absorbing set as the deterministic case as in e.g. [14]; on the other hand, owing to the influences of stochastic driving system, one can not obtain the fixed invariant set for stochastic dynamical system in general.

In order to overcome these drawbacks, Flandoli etc. in [9,10] introduce the theory of pathwise random dynamical systems and (pathwise) random attractors for the autonomous stochastic equations, in which the random attractor is a family of compact sets depending on random parameters and has some invariant property under the action of the random dynamical system. Recent theory in [12,27] are related to non-autonomous pathwise random dynamical systems and pullback random attractors for non-autonomous stochastic equations, where the pullback random attractor is a family of compact sets depending on both random parameters and deterministic time symbols. Up to now, there are many results on the existence and uniqueness of random attractor, one can refer to [16,20,29,36] for the autonomous stochastic equations and [22,28,29,30] for the non-autonomous stochastic equations. In addition, for the result about random attractors for equation (1) with linear noise, see, e.g. [5,13,16,29,36].

However, when one investigates the dynamics of stochastic evolution equations driven by nonlinear noise, the existence of random attractors can not be established directly, since the serious challenge is that the existence of random dynamical system is unknown in general for these kinds of systems. As far as it is known, up to now there are two ways to over come this difficulty in some sense. One method is to investigate the dynamic behavior of the Wong-Zaki approximate system corresponding to original equation. For example, Lu and Wang in [21] get the existence of pullback random attractor for the Wong-Zakai approximate system of a stochastic reaction-diffusion equation with the nonlinear noise in some bounded spatial domain, and later, Wang et al. in [34] extend the result of [21] to unbounded domains by using the method of tail estimates. The another method is established by Kloeden et al. in [18] and Wang in [31], that is, they extend the concept of pathwise random attractor to mean context and establish the corresponding existence theory of mean random attractor for random dynamical system. There are some relevant works, see e.g. [32,33].

The first purpose of this article is to establish the existence of weak pullback mean random attractors for Eq. (1) by using the theory of [31]. Toward this end, we first need to get the existence and uniqueness of solution for Eq. (1). Unlike reference [31], the existence of solution for Eq. (1) can not be obtained directly by using the abstract result (Theorem 4.2.4) in [24] since the drift term

f (x, u)

is allowed to be polynomial growth of arbitrary order with respect to u in this article. We aim to prove the existence and uniqueness of the solution for Eq. (1) by using the approach of [32], in which the author prove existence of solutions for a stochastic reaction-diffusion equations involving drift term

f (x, t, u)

with polynomial growth of any order and nonlinear diffusion term

σ (t, u)

, and the embedding

H^{k} (R^{N}) ↪ L^{p} (R^{N})

for

2 \leq p \leq \frac{2 N}{N - 2 k}

(

N \geq 2 k

) plays an essential role in this proof. Hence, we show the embedding result of the corresponding Sobolev space with weight

σ (x)

in Section 2. In Section 3, we show the solution generate a mean random dynamical system and establish the existence of weak pullback random attractors for Eq. (1). We shall remark that since the mean random dynamical system is defined on the Banach space

L^{p} (Ω, X)

consisting of all Bochner integrable functions and corresponding probability space

(Ω, F, P)

lacks some topological structure, we only get the weakly compact property and weakly attracting property of mean random attractors for (1) in

L^{2} (Ω, X)

The second goal is to investigate dynamic behavior of the Wong-Zakai approximate system for Eq. (1). We prove the existence of pullback random attractor for the Wong-Zakai approximate system for equation (1) with nonlinear diffusion term

h (t, x, u)

, which is allowed to be polynomial growth, and we also show that the pullback random attractor of Wong-Zaki approximation for Eq. (1) converges to the attractor of Eq. (1) as the size of approximation tends to zero, when

h (t, x, u)

is equal to u. This work will be done in section 4. We remark that when we prove the pullback asymptotic compactness, we use method of weighted sobolev spaces to overcome the non-compactness of usual Sobolev embeddings in the case of unbounded domain, which is different from that of [21].

In what follows of this article, the constant C represents some positive constant and may change from line to line.

2. Preliminaries

2.1. Functional setting

In this subsection, we introduce some function spaces and present some embedding results, which will be used in our proof.

Throughout this article, we let

(X, ∥ \cdot ∥_{X})

be a separable Banach space and

L^{p} (Ω, F; X)

(

1 < p < \infty

) be the Banach space consisting of all strongly measurable and Bochner integrable functions

Ψ

from

Ω

to X such that

\begin{matrix} {∥ Ψ ∥}_{L^{p} (Ω, F; X)} = (\int_{Ω} {∥ Ψ ∥}_{X}^{p} {d p)}^{\frac{1}{p}} < + \infty . \end{matrix}

(2)

Denote by

(Ω, F, {F_{t}}_{t \in R}, P)

the complete filtered probability space satisfying the usual condition, i.e.,

{F_{t}}_{t \in R}

is an increasing right continuous family of sub-

σ

-algebras of

F

that contains all

P

-null sets. We use

L^{p} (Ω, F_{t}; X)

to represent the subspace of

L^{p} (Ω, F; X)

, which consists of all functions belonging to

L^{p} (Ω, F; X)

and being strongly

F_{t}

-measurable. For simplicity of notation, we denote by

∥ \cdot ∥

the norm in

L^{2} (O)

and

L^{2} (Ω, F_{t}; L^{2} (O))

To investigate Eq. (1), we introduce the weighted Sobolev space

D_{0}^{1, 2} (O, σ)

defined by the completion of

C_{0}^{\infty} (O)

with norm

{∥ \cdot ∥}_{D_{0}^{1, 2} (O, σ)}

\begin{matrix} {∥ u ∥}_{D_{0}^{1, 2} (O, σ)} : = (\int_{O} {σ (x) | \nabla u |}^{2} {d x)}^{\frac{1}{2}} . \end{matrix}

(3)

And one can easily check that

D_{0}^{1, 2} (O, σ)

is a Hilbert space with the inner product

{(\cdot, \cdot)}_{σ}

\begin{matrix} {(u, v)}_{σ} : = \int_{O} σ (x) \nabla u \nabla v d x . \end{matrix}

(4)

If condition

(H_{α})

(or

(H_{α, β})

on unbounded domain) holds, the operator

A = - div (σ (x) \nabla u)

is positive and self-adjoint with domain defined by

D (A) : = {u \in D_{0}^{1, 2} (O, σ) : A u \in L^{2} (O)} .

Furthermore, one can easily observe that if

σ

satisfies

(H_{α})

and

(H_{α, β})

, then there exists a finite set

A = {a_{1}, a_{2}, \dots, a_{k}} \subseteq \bar{O}

and

δ, r > 0

such that the balls

B_{i} = B_{r} (a_{i})

i = 1, 2, \dots, k

, are disjoint and

\begin{matrix} σ (x) & \geq δ | x - a_{i} |^{α} for x \in B_{i} \cap Ω, i = 1, 2, \dots, k, \end{matrix}

(5)

\begin{matrix} σ (x) & \geq δ for x \in Ω ∖ \cup_{i} B_{i}, \end{matrix}

(6)

and moreover, if

Ω

is unbounded, then there exists

R > 0

such that

\begin{matrix} σ (x) \geq {δ | x |}^{β} for x \in Ω, | x | > R . \end{matrix}

(7)

The following spaces will also be needed:

$D^{p} (A) : = {u \in D_{0}^{1, 2} (O, σ) : A u \in L^{p} (O)}$ ;
$D_{0}^{- 1} (O, σ)$ := the dual space of $D_{0}^{1, 2} (O, σ)$ ;
$H_{0}^{m} (O, σ)$ := the closure of $C_{0}^{\infty} (O)$ with norm ${∥ \cdot ∥}_{H^{m} (O, σ)}$ , defined by

$\begin{matrix} {∥ u ∥}_{H^{m} (O, σ)}^{2} : = \sum_{1 \leq | κ | \leq m} \int_{O} σ (x) | D^{κ} {u |}^{2} d x + \int_{O} {| u |}^{2} d x, m \in N^{+}, \end{matrix}$

where $κ = (κ_{1}, κ_{2}, \dots, κ_{N})$ is a multi-index of order $| κ | = κ_{1} + κ_{2} + \dots + κ_{N}$ .

Lemma 1.

([19]) There exists a constant

c_{1}

such that the following inequality holds true for all

u \in C_{0}^{\infty} (R^{N})

\begin{matrix} {(\int_{R^{N}} {| u |}^{2_{α}^{*}} d x)}^{\frac{1}{2_{α}^{*}}} \leq c_{1} {(\sum_{| κ | = m} \int_{R^{N}} {| x |}^{α} {| D^{κ} u |}^{2} d x)}^{\frac{1}{2}}, \end{matrix}

where

2_{α}^{*} = \frac{2 N}{N + 2 α - 2 m}

with

N - 2 m \geq 0

Lemma 2.

Let

σ (x)

satisfy assumption(

H_{α}

)(or(

H_{α, β}

)on unbounded domain). Then there exists a constant

c_{2}

such that

{(\int_{O} {| u |}^{2_{α}^{*}} d x)}^{\frac{1}{2_{α}^{*}}} \leq c_{2} {(\sum_{| κ | = m} \int_{O} σ (x) {| D^{κ} u |}^{2} d x)}^{\frac{1}{2}}, f o r e v e r y u \in C_{0}^{\infty} (O) .

Proof.

By using Lemma 1, the Rellich-Kondrachov Theorem, and the General Sobolev inequality, we can get the conclusion of Lemma 2 in the similar way as in the proof of Proposition 2.5 in [8]. We omit the process here. □

The following embedding results play an important role in our proof in Section 3 and Section 4.

Lemma 3. ([8])Let

σ (x)

satisfies assumption(

H_{α}

)(or(

H_{α, β}

)on unbounded domain). Then it holds the compact embedding

D_{0}^{1, 2} (O, σ) ↪ ↪ L^{2} (O)

Lemma 4.

Let

σ (x)

satisfies assumption(

H_{α}

)(or(

H_{α, β}

)on unbounded domain). Then it holds the continuous embedding

H_{0}^{m} (O, σ) ↪ L^{p} (O), f o r 2 \leq p \leq 2_{α}^{*} .

Proof.

Note that

2_{α}^{*} > 2

for

α \in (0, 1)

. Then we can get by the interpolation theorem and Lemma 2 that

{∥ u ∥}_{L^{p} (O)} \leq {C ∥ u ∥}^{θ} {∥ u ∥}_{L^{2_{α}^{*}} (O)}^{1 - θ} \leq C {∥ u ∥}_{H^{m} (O, σ)}, f o r a n y u \in H_{0}^{m} (O, σ),

where

θ = \frac{2 (2_{α}^{*} - p)}{p (2_{α}^{*} - 2)}

. The proof is completed. □

2.2. Theory of Random Attractors

In this subsection, we introduce some definitions and known results about weak pullback mean random attractors and pullback random attractors.

Definition 1.

A family of mappings

Φ = {Φ (t, τ) : t \in R^{+}, τ \in R}

is called mean random dynamical system on

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

if the following conditions hold for all

τ \in R

and

t, s \in R^{+}

(i): $Φ (t, τ)$ maps $L^{p} (Ω, F_{τ}, X)$ to $L^{p} (Ω, F_{t + τ}, X)$ ;
(ii): $Φ (0, τ)$ is the identity operator on $L^{p} (Ω, F_{τ}, X)$ ;
(iii): $Φ (t + s, τ) = Φ (t, τ + s) \circ Φ (s, τ)$ .

Let

D = {D (τ) \subseteq L^{p} (Ω, F_{τ}; X) : τ \in R}

be a family of nonempty bounded sets and

D_{0}

be a collection of such families satisfying some conditions. The collection

D_{0}

is said to be inclusion-closed if

D = {D (τ) : τ \in R} \in D_{0}

, then every family

O = {O (τ) : O (τ) \subseteq D (τ), τ \in R} \in D_{0}

Definition 2.

A family of sets

K = {K (τ) : τ \in R} \in D_{0}

is called a

D_{0}

-pullback absorbing set for

Φ

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

if for every

τ \in R

and

D \in D_{0}

, there exists

T = T (τ, D) > 0

such that

\begin{matrix} Φ (t, τ - t, D (τ - t)) \subseteq K (τ), \forall t \geq T . \end{matrix}

Moreover, if

K (τ)

is a weakly compact nonempty subset of

L^{p} (Ω, F_{τ}; X)

for each

τ \in R

, then

K = {K (τ) : τ \in R}

is said to be a weakly compact

D_{0}

-pullback absorbing set for

Φ

Definition 3.

A family of sets

K = {K (τ) : τ \in R} \in D_{0}

is said to be a

D_{0}

-pullback weakly attracting set of

Φ

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

if for each

τ \in R, D \in D_{0}

and every weak neighborhood

N^{w} (K (τ))

K (τ)

L^{p} (Ω, F_{τ}; X)

, there exists some

T = T (τ, D, N^{w} (K (τ))) > 0

such that

\begin{matrix} Φ (t, τ - t, D (τ - t)) \subseteq N^{w} (K (τ)), \forall t \geq T . \end{matrix}

Definition 4.

We say a family

A = {A (τ) : τ \in R} \in D_{0}

is a weak

D_{0}

-pullback mean random attractor for

Φ

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

if it satisfies the following properties:

Weak compactness: for any $τ \in R$ , $A (τ)$ is a weakly compact subset of $L^{p} (Ω, F_{τ}; X)$ .
Pullback weak attraction: for any $τ \in R$ , $A (τ)$ is a $D_{0}$ -pullback weakly attracting set of $Φ$ .
Minimality: for any $τ \in R$ , the family $A$ is the minimal element of $D_{0}$ in the sense that if $B = {B (τ) : τ \in R} \in D_{0}$ is another weakly compact $D_{0}$ -pullback weakly attracting set of $Φ$ , then $A (τ) \subseteq B (τ)$ .

The following result about the existence and uniqueness of weak

D_{0}

-pullback mean random attractors for

Φ

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

comes from [31].

Lemma 5.

Suppose that

D_{0}

is an inclusion-closed collection of some families of nonempty bounded subsets of

L^{p} (Ω, F; X)

and Φ is a weak mean random dynamical system on

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

. If Φ possesses a weakly compact

D_{0}

-pullback absorbing set

K \in D_{0}

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

, then Φ possesses a unique weak

D_{0}

-pullback mean random attractor

A \in D_{0}

L^{p} (Ω, F; X)

over

(Ω, F, {F_{t}}_{t \in R}, P)

, which is given by

\begin{matrix} A (τ) = Ω^{w} (K, τ) = ⋂_{r \geq 0} {\bar{⋃_{t \geq r} Φ (t, τ - t, K (τ - t))}}^{w}, \forall τ \in R, \end{matrix}

where the closure is taken with respect to the weak topology of

L^{p} (Ω, F_{τ}; X)

Denote by

D = {D (τ, ω) : τ \in R, ω \in Ω}

a family of nonempty bounded subsets of X and

D_{1}

a collection of such families satisfying some conditions. Let

(Ω, F, P, {θ_{t}}_{t \in R})

be a metric dynamical system. We now introduce the pathwise random dynamical systerm as in [6,9,27].

Definition 5.

A mapping

Ψ : R^{+} \times R \times Ω \times X \mapsto X

is said to be a continuous pathwise random dynamical system (or a continuous cocycle) on X over

(Ω, F, P, {θ_{t}}_{t \in R})

if the following conditions hold for all

τ \in R

ω \in Ω

and

t, s \in R^{+}

(i): $Ψ (\cdot, τ, \cdot, \cdot) : R^{+} \times Ω \times X \mapsto X$ is $(B (R^{+}) \times F \times B (X), B (X))$ -measurable;
(ii): $Ψ (0, τ, ω, \cdot)$ is the identity operator on X;
(iii): $Ψ (t + s, τ, ω, \cdot) = Ψ (t, τ + s, θ_{s} ω, \cdot) \circ Ψ (s, τ,, ω, \cdot)$ ;
(iv): $Ψ (t, τ, ω, \cdot) : X \mapsto X$ is continuous.

Definition 6.

A family

K = {K (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

is said to be a

D_{1}

-pullback absorbing set for a cocycle

Ψ

if for every

τ \in R, ω \in Ω

and

D \in D_{1}

, there exists some

T = T (τ, D, ω) > 0

such that

\begin{matrix} Ψ (t, τ - t, θ_{- t} ω, D (τ - t, θ_{- t} ω)) \subseteq K (τ, ω) for all t \geq T . \end{matrix}

Moreover, If for every

τ \in R

and

ω \in Ω, K (τ, ω)

is a closed nonempty subset of X and is measurable in

ω

with respect to

F

, then K is said to be a closed measurable

D_{1}

-pullback absorbing set for

Ψ

Definition 7.

We say that cocycle

Ψ

D_{1}

-pullback asymptotically compact in X if for every

τ \in R

and

ω \in Ω

, the sequence

\begin{matrix} {Ψ (t_{n}, τ - t_{n}, θ_{- t_{n}} ω, x_{n})}_{n = 1}^{\infty} has a convergent subsequence in X, \end{matrix}

t_{n} \to + \infty,

and

x_{n} \in B (τ - t_{n}, θ_{- t_{n}} ω)

with

{B (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

Definition 8.

A family

A = {A (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

is said to be a

D_{1}

-pullback random attractor for

Ψ

if the following properties hold for all

τ \in R

and

ω \in Ω

(i): Measurability and Compactness: $A$ is measurable in $ω$ with respect to $F$ and $A (τ, ω)$ is compact in X;
(ii): Invariance: $A$ is invariant in the sense that $Ψ (t, τ, ω, A (τ, ω)) = A (τ + t, θ_{t} ω)$ , $\forall t \geq 0$ ;
(iii): Pullback attracting: $A$ attracts $D_{1}$ in the sense that for any $D \in D_{1}$ ,

$\begin{matrix} lim_{t \to + \infty} {dist}_{X} (Ψ (t, τ - t, θ_{- t} ω, D (τ - t, θ_{- t} ω)), A (τ, ω)) = 0, \end{matrix}$

where ${dist}_{X}$ is the Hausdorff semi-distance in X.

3. Mean Random Attractors for Stochastic Semi-linear Degenerate Parabolic Equation

Let U be a separable Hilbert space and

L_{2} (U, L^{2} (O))

be the Hilbert space consisting of all Hilbert-Schmidt operators from U to

L^{2} (O)

with norm

{∥ \cdot ∥}_{L_{2} (U, L^{2} (O))}

. We consider the following non-autonomous stochastic semi-linear degenerate parabolic equation defined on any bounded or unbounded domain

O \subseteq R^{N}

\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - div (σ (x) \nabla u) + λ u + f (x, u) = g (t, x) + h (t, u) \frac{d W}{d t}, t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (t, x) |}_{\partial O} = 0, t > τ, \end{matrix} \end{matrix}

(8)

where W is a two-sided U-valued cylindrical Wiener process defined on the complete filtered probability space

(Ω, F, {F_{t}}_{t \in R}, P)

, while

σ (x)

λ

and

g (t, x)

are the same as described in Section 1. In this section, the stochastic term in Eq. (8) is understood in the sense of Itô integration. Since the Itô integral is martingale, it is convenient for us to take expectation and get the existence of weak pullback mean random attractor.

Let

O

be a bounded domain (or an unbounded domain) and let the nonnegative function

σ (x)

satisfy

(H_{α})

(or (

H_{α, β}

)). We assume that

f : O \times R \mapsto R

is a smooth nonlinear function such that

f (x, 0) = 0

and for all

x \in O

and

u \in R

\begin{matrix} \frac{\partial f}{\partial u} (x, u) \geq - ϕ_{1} (x), \end{matrix}

(9)

\begin{matrix} f (x, u) u \geq a_{1} {| u |}^{p} - ϕ_{2} (x), \end{matrix}

(10)

\begin{matrix} | f (x, u) | \leq a_{2} {| u |}^{p - 1} + ϕ_{3} (x), \end{matrix}

(11)

where

a_{1}, a_{2}, a_{3}, p > 2

are positive constants, and

ϕ_{1} (x) \in L^{\infty} (O)

with

ϕ_{1} (x) \geq 0

ϕ_{2} (x) \in L^{1} (O)

ϕ_{3} (x) \in L^{p_{1}} (O)

with

\frac{1}{p} + \frac{1}{p_{1}} = 1

. We also assume

f (x, u)

is locally Lipschitz continuous in u, i.e., for each bounded interval

I \subseteq R

, there is

a_{I} > 0

such that

\begin{matrix} | f (x, u_{1}) - f (x, u_{2}) | \leq a_{I} | u_{1} - u_{2} |, \forall x \in O, u_{1}, u_{2} \in I . \end{matrix}

(12)

Assume

h : R \times Ω \times L^{2} (O) \mapsto L_{2} (U, L^{2} (O))

satisfies the following conditions:

(A₁): For any $t \in R$ , $ω \in Ω$ and $u \in L^{2} (O)$ , there are positive constants $a_{3} < \frac{1}{2} λ$ and L such that

$\begin{matrix} {∥ h (t, ω, u) ∥}_{L_{2} (U, L^{2} (O))}^{2} \leq a_{3} {∥ u ∥}^{2} + L . \end{matrix}$

(13)
(A₂): For each $r > 0$ , there is a positive constant $a_{r}$ depending on r such that for every $t \in R$ , $ω \in Ω$ , and $u, v \in L^{2} (O)$ with $∥ u ∥ \leq r$ and $∥ v ∥ \leq r$ ,

$\begin{matrix} {∥ h (t, ω, u) - h (t, ω, v) ∥}_{L_{2} (U, L^{2} (O))}^{2} \leq a_{r} {∥ u - v ∥}^{2} . \end{matrix}$

(14)

Moreover, we suppose that for each given

u \in L^{2} (O)

σ (\cdot, \cdot, u) : R \times Ω \mapsto L_{2} (U, L^{2} (O))

is progressively measurable.

We now show the solution of Eq. (8) can define a mean random dynamical system. The definition of solution for Eq. (8) is given as follows in this case.

Definition 9.

Let

u_{τ} \in L^{2} (Ω, F_{τ}; L^{2} (O))

and

T > τ

. A

L^{2} (O)

-valued

F_{t}

-adapted stochastic process u is called a solution of (8) on

[τ, T]

with initial data

u_{τ}

u \in L^{2} (Ω, C ([τ, T]; L^{2} (O))) \cap L^{2} (Ω \times [τ, T]; D_{0}^{1, 2} (O, σ)) \cap L^{p} (Ω \times [τ, T]; L^{p} (O))

and P-a.s. satisfies

\begin{matrix} (u (t), & ζ) + \int_{τ}^{t} (σ (x) \nabla u, \nabla ζ) d s + λ \int_{τ}^{t} (u, ζ) d s + \int_{τ}^{t} \int_{O} f (u) ζ d x d s = \int_{τ}^{t} (g (s), ζ) d s \\ + \int_{τ}^{t} (h (s, u) d W (s), ζ), \forall t \in [τ, T], ζ \in D_{0}^{1, 2} (O, σ) \cap L^{p} (O) . \end{matrix}

Using Lemma 3, Lemma 4, we can get the following result in a similar way that have been used in [32].

Lemma 6.

Let

T > τ

and

u_{τ} \in L^{2} (Ω, F_{τ}; L^{2} (O))

. If conditions (9)-(14) hold, then there exists a unique solution to Eq. (8) in the sense of Definition 9. Besides,

\begin{matrix} E (sup_{t \in [τ, T]} ∥ u (t) ∥^{2}) < \infty . \end{matrix}

(15)

Note that

u \in L^{2} (Ω, C ([τ, T]; L^{2} (O)))

for all

T > τ

, which implies that

u \in C ([τ, \infty); L^{2} (Ω, L^{2} (O)))

. Thus we can define the mean random dynamical system

Φ

for Eq. (8) on

L^{2} (Ω, F; L^{2} (O))

\begin{matrix} Φ (t, τ, u_{τ}) = u (t + τ, τ, u_{τ}), t > 0, τ \in R, \end{matrix}

where

u_{τ} \in L^{2} (Ω, F_{τ}; L^{2} (O))

and u is the solution of system (8) with initial data

u_{τ}

Let

D = {D (τ) \subseteq L^{2} (Ω, F_{τ}; L^{2} (O)) : τ \in R}

be a family of nonempty bounded sets. A family

D

is said to be tempered if for any

ν > 0

, there is

\begin{matrix} lim_{τ \to - \infty} e^{ν τ} sup_{u \in D (τ)} {∥ u ∥}^{2} = 0 . \end{matrix}

(16)

We denote by

D_{0}

the collection of all tempered families of nonempty bounded subsets of

L^{2} (Ω, F_{τ}; L^{2} (O))

, that is,

D_{0} = {D = {D (τ) \subseteq L^{p} (Ω, F_{τ}; L^{2} (O)) : D (τ) \neq \emptyset, bounded,, τ \in R} : D satisfies (16)} .

From now on, we assume:

\begin{matrix} \int_{- \infty}^{τ} e^{λ s} {∥ g (s, \cdot) ∥}^{2} d s < + \infty, \forall τ \in R . \end{matrix}

(17)

To get the existence of tempered random attractors, we further assume:

\begin{matrix} lim_{τ \to - \infty} e^{ν τ} \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ, \cdot) ∥}^{2} d s = 0, \forall ν > 0 . \end{matrix}

(18)

To investigate the existence of weak

D_{0}

-pullback mean random attractors for Eq. (8), we need the uniform estimate of solutions, and by the following result, we can construct a weakly compact

D_{0}

-pullback absorbing set for

Φ

Lemma 7.

Suppose (9)-(14) and (17) hold. Then for every

τ \in R

and

D \in D_{0}

, there exists some

T = T (τ, D) > 0

such that for all

t \geq T

and

u_{τ - t} \in D (τ - t)

, the solution u to Eq. (8) satisfies

\begin{matrix} E (∥ u (τ, τ - t, u_{τ - t}) ∥^{2}) \leq M + M \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ) ∥}^{2} d s, \end{matrix}

(19)

where M is a positive constant independent of τ and

D

Proof.

By the Itô formula, we obtain from (8) that for each

r \geq τ - t

\begin{matrix} ∥ u (r, τ - t, u_{τ - t}) ∥^{2} + 2 \int_{τ - t}^{r} ∥ u (s, τ - t, u_{τ - t}) ∥_{D_{0}^{1, 2} (O, σ)}^{2} d s + 2 λ \int_{τ - t}^{r} {∥ u (s, τ - t, u_{τ - t}) ∥}^{2} d s \\ + 2 \int_{τ - t}^{r} \int_{O} f (x, u (s, τ - t, u_{τ - t})) u (s, τ - t, u_{τ - t}) d x d s \\ = & ∥ u_{τ - t} ∥^{2} + 2 \int_{τ - t}^{r} (g (s), u (s, τ - t, u_{τ - t})) d s + \int_{τ - t}^{r} {∥ h (s, u (s, τ - t, u_{τ - t})) ∥}_{L_{2} (U, L^{2} (O))}^{2} d s \\ + 2 \int_{τ - t}^{r} (u (s, τ - t, u_{τ - t}), h (s, u (s, τ - t, u_{τ - t})) d W (s)), \end{matrix}

(20)

Taking the expectation on both sides of (20), we get, for almost all

r \geq τ - t

, that

\begin{matrix} E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) + 2 \int_{τ - t}^{r} E (∥ u (s, τ - t, u_{τ - t}) ∥_{D_{0}^{1, 2} (O, σ)}^{2}) d s \\ + 2 λ \int_{τ - t}^{r} E (∥ u (s, τ - t, u_{τ - t}) ∥^{2}) d s + 2 \int_{τ - t}^{r} E (\int_{O} f (x, u (s, τ - t, u_{τ - t})) u (s, τ - t, u_{τ - t}) d x) d s \\ = & E (∥ u_{τ - t} ∥^{2}) + 2 \int_{τ - t}^{r} E (g (s), u (s, τ - t, u_{τ - t})) d s \\ + \int_{τ - t}^{r} E (∥ h (s, u (s, τ - t, u_{τ - t})) ∥_{L_{2} (U, L^{2} (O))}^{2}) d s . \end{matrix}

(21)

Thus, for almost all

r \geq τ - t

, we have

\begin{matrix} \frac{d}{d r} & E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) + 2 E (∥ u (r, τ - t, u_{τ - t}) ∥_{D_{0}^{1, 2} (O, σ)}^{2}) \\ + & 2 λ E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) + 2 E (\int_{O} f (x, u (r, τ - t, u_{τ - t})) u (r, τ - t, u_{τ - t}) d x) \\ = 2 & E (g (r), u (r, τ - t, u_{τ - t})) \\ + & E (∥ h (r, u (r, τ - t, u_{τ - t})) ∥_{L_{2} (U, L^{2} (O))}^{2}) . \end{matrix}

(22)

Now, we estimate each item on the right-hand side of (22). By (10) we have that

\begin{matrix} \int_{O} f (u (r, τ - t, u_{τ - t})) u (r, τ - t, u_{τ - t}) d x \\ \geq & a_{1} \int_{O} | u (r, τ - t, u_{τ - t}) |^{p} d x - {∥ ϕ_{2} ∥}_{L^{1} (O)}, \end{matrix}

(23)

which implies

\begin{matrix} 2 E (\int_{O} f (u (r, τ - t, u_{τ - t})) u (r, τ - t, u_{τ - t}) d x) \\ \geq & 2 a_{1} E (∥ u (r, τ - t, u_{τ - t}) ∥_{L^{p} (O)}^{p}) - 2 ∥ ϕ_{2} ∥_{L^{1} (O)} . \end{matrix}

(24)

Note that

\begin{matrix} (g (r), u (r, τ - t, u_{τ - t})) \\ \leq & \frac{λ}{4} ∥ u (r, τ - t, u_{τ - t}) ∥^{2} + \frac{1}{λ} {∥ g (r) ∥}^{2}, \end{matrix}

(25)

which implies that

\begin{matrix} 2 E (g (r), u (r, τ - t, u_{τ - t})) \\ \leq & \frac{λ}{2} E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) + \frac{2}{λ} {∥ g (r) ∥}^{2} . \end{matrix}

(26)

We deduce from (22)-(26) and (13) that, for almost all

r \geq τ - t

\begin{matrix} \frac{d}{d r} & E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) + λ E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) \\ \leq & \frac{2}{λ} {∥ g (r) ∥}^{2} + 2 {∥ ϕ_{2} ∥}_{L^{1} (O)} + L . \end{matrix}

(27)

Applying Gronwall’s inequality to (27), we get

\begin{matrix} E (∥ u (r, τ - t, u_{τ - t}) ∥^{2}) \\ \leq & e^{λ (τ - t - r)} E (∥ u_{τ - t}) ∥^{2}) + \frac{1}{λ} (2 ∥ ϕ_{2} ∥_{L^{1} (O)} + L) + e^{- λ r} \int_{τ - t}^{r} e^{λ s} \frac{2}{λ} {∥ g (s) ∥}^{2} d s . \end{matrix}

(28)

Then we get that

\begin{matrix} E (∥ u (τ, τ - t, u_{τ - t}) ∥^{2}) \\ \leq & e^{- λ t} E (∥ u_{τ - t} ∥^{2}) + \frac{1}{λ} (2 ∥ ϕ_{2} ∥_{L^{1} (O)} + L) + e^{- λ τ} \frac{2}{λ} \int_{- \infty}^{τ} e^{λ s} {∥ g (s) ∥}^{2} d s . \end{matrix}

(29)

Since

u_{τ - t} \in D (τ - t)

and

D = {D (τ) : τ \in R} \in D_{0}

, we get

\begin{matrix} e^{- λ τ} e^{λ (τ - t)} E (∥ u_{τ - t} ∥^{2}) \to 0 a s t \to + \infty . \end{matrix}

Therefore, there exists

T = T (τ, D) > 0

such that for all

t \geq T

\begin{matrix} e^{- λ t} E (∥ u_{τ - t} ∥^{2}) \leq 1 . \end{matrix}

(30)

By (29) and (30), we get, for all

t \geq T

, there exists some positive constant M independent of

τ

and

D

such that

\begin{matrix} E (∥ u (τ, τ - t, u_{τ - t}) ∥^{2}) \leq M + M \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ) ∥}^{2} d s . \end{matrix}

This completes the proof. □

Corollary 1.

Let (9)-(14) and (17)-(18) hold. Then the mean random dynamical system Φ for Eq. (8) possesses a weakly compact

D_{0}

-pullback absorbing set

K_{0} = {K_{0} (τ) : τ \in R} \in D_{0},

which is given by,

\begin{matrix} K_{0} (τ) = {u \in L^{2} (Ω, F_{τ}; L^{2} (O)) {: E (∥ u ∥}^{2}) \leq R_{0} (τ)}, \end{matrix}

(31)

where

\begin{matrix} R_{0} (τ) : = M + M \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ) ∥}^{2} d s \end{matrix}

(32)

with M being the same constant as in Lemma 7.

Proof.

We know that for each

τ \in R, K_{0} (τ)

in (31) is a bounded and closed convex subset of

L^{2} (Ω, F_{τ}; L^{2} (O))

, and therefore it is weakly compact in

L^{2} (Ω, F_{τ}; L^{2} (O))

. Lemma 7 indicates that for every

τ \in R

and

D \in D_{0}

, there exists

T = T (τ, D) > 0

such that

\begin{matrix} Φ (t, τ - t, D (τ - t)) \subseteq K_{0} (τ), \forall t \geq T . \end{matrix}

(33)

In addition, from (18) and (32), we get for any

ν > 0

\begin{matrix} lim_{τ \to - \infty} e^{ν τ} sup_{u \in K_{0} (τ)} ∥ u ∥ = 0, \end{matrix}

that is

K_{0} \in D_{0}

. Hence,

K_{0}

is a weakly compact

D_{0}

-pullback absorbing set for

Φ

. □

Theorem 1.

Suppose (9)-(14) and (17)-(18) hold. Then the mean random dynamical system Φ for problem (8) possesses a unique weak

D_{0}

-pullback mean random attractor

{\bar{A}}_{0} = {{\bar{A}}_{0} (τ) : τ \in R} \in D_{0}

L^{2} (Ω, F; L^{2} (O))

over

(Ω, F, {F_{t}}_{t \in R}, P)

Proof.

From Lemma 5 and Corollary 1, we can easily get the existence and uniqueness of weak

D_{0}

-pullback mean random attractor

{\bar{A}}_{0} \in D_{0}

Φ

for Eq. (8). □

4. Wong-ZaKai Approximations of Stochastic Semi-linear Degenerate Parabolic Equation

In this section, we consider the following stochastic semi-linear degenerate parabolic equation

\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - div (σ (x) \nabla u) + λ u + f (x, u) = g (t, x) + h (t, x, u) \circ \frac{d W}{d t}, t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (x, t) |}_{\partial O} = 0, t > τ . \end{matrix} \end{matrix}

(34)

Here

W = ω (t)

is a two-sided real-valued Wiener process on a probability space and the other terms are the same as described in section 1. The symbol

‘ \circ ’

indicates that the stochastic term in Eq. (34) is understood in the sense of Stratonovich’s integration.

We remark that, in this section, we consider the stochastic term of Eq. (34) in the sense of Stratonovich’s integration because the Stratonovich’s interpretation is more appropriate than Itô’s when we consider the pathwise dynamical behavior (fixed any

ω \in Ω

) of the Wong-Zakai approximate system corresponding to the equation (see [35] for details).

4.1. Random dynamical systems for Wong-Zakai Approximations

In this subsection, we first define a continuous cocycle

Ψ

for Wong-Zakai approximate system of Eq. (34), and then prove that there exists a unique pullback random attractor for the cocycle

Ψ

Let

O

be a bounded domain (or an unbounded domain) and let the nonnegative function

σ (x)

satisfy

(H_{α})

(or (

H_{α, β}

)). In what follows, We assume that

f : O \times R \mapsto R

is a smooth nonlinear function such that for all

x \in O

and

u \in R

\begin{matrix} f (x, u) u \geq α_{1} {| u |}^{p} - β_{1} (x), \end{matrix}

(35)

\begin{matrix} | f (x, u) | \leq α_{2} {| u |}^{p - 1} + β_{2} (x), \end{matrix}

(36)

\begin{matrix} \frac{\partial f (x, u)}{\partial u} \geq α_{3} {| u |}^{p - 2} - β_{3} (x), \end{matrix}

(37)

where

p > 2

α_{1}, α_{2}, α_{3}

are positive numbers,

β_{1} (x) \in L^{1} (O), β_{2} (x) \in L^{p_{1}} (O)

with

\frac{1}{p_{1}} + \frac{1}{p} = 1

β_{3} (x) \in L^{\infty} (O)

. Let h be a continuous function and for all

t, u \in R

x \in O

, satisfy

\begin{matrix} | h (t, x, u) | \leq ψ_{1} (t, x) {| u |}^{q - 1} + ψ_{2} (t, x), \end{matrix}

(38)

\begin{matrix} | \frac{\partial}{\partial u} h (t, x, u) | \leq ψ_{3} (t, x) {| u |}^{q - 2} + ψ_{4} (t, x), \end{matrix}

(39)

where

2 \leq q < p, ψ_{1} \in L_{loc}^{\frac{p}{p - q}} (R; L_{loc}^{\frac{p}{p - q}} (O))

and

ψ_{2} \in L_{loc}^{p_{1}} (R; L^{p_{1}} (O))

, and

ψ_{3}, ψ_{4} \in L^{\infty} (R; L^{\infty} (O))

In the sequel, let

(Ω, F, P)

be the classical Wiener probability space, where

\begin{matrix} Ω = C_{0} (R, R) : = {ω \in C (R, R) : ω (0) = 0} \end{matrix}

(40)

with the open compact topology. The Brownian motion has the form

W (t, ω) = ω (t)

. Consider the Wiener shift

θ_{t}

on the probability space

(Ω, F, P)

defined by

\begin{matrix} θ_{t} ω (\cdot) = ω (t + \cdot) - ω (t) . \end{matrix}

(41)

Then from [1], we get that

(Ω, F, P, {θ_{t}}_{t \in R})

is a metric dynamical system and there exists a

{θ_{t}}_{t \in R}

-invariant subset

\tilde{Ω} \subseteq Ω

of full

P

measure such that for each

ω \in \tilde{Ω}

\begin{matrix} \frac{ω (t)}{t} \to 0 as t \to \pm \infty . \end{matrix}

(42)

For brevity, we identify the space

\tilde{Ω}

with

Ω

. For any given

δ \neq 0

, define the random variable

G_{δ}

\begin{matrix} G_{δ} (ω) = \frac{ω (δ)}{δ}, \forall ω \in Ω . \end{matrix}

(43)

We get from (41) and (43) that

\begin{matrix} G_{δ} (θ_{t} ω) = \frac{ω (t + δ) - ω (t)}{δ} and \int_{0}^{t} G_{δ} (θ_{s} ω) d s = \int_{t}^{t + δ} \frac{ω (s)}{δ} d s + \int_{δ}^{0} \frac{ω (s)}{δ} d s . \end{matrix}

(44)

By the continuity of

ω

and (44), the following result have been proved in [21].

Lemma 8.

Let

τ \in R

T > 0

, and

ω \in Ω

. Then for each

ϵ > 0

, there is a constant

δ^{'} = δ^{'} (ϵ, τ, ω, T) > 0

such that for every

0 < | δ | < δ^{'}

and

t \in [τ, τ + T]

\begin{matrix} | \int_{0}^{t} G_{δ} (θ_{s} ω) d s - ω (t) | < ϵ . \end{matrix}

(45)

Let’s consider the Wong-Zakai approximate system of Eq. (34):

\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} + (- div (σ (x) \nabla u)) + λ u + f (x, u) = g (t, x) + h (t, x, u) G_{δ} (θ_{t} ω), t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (x, t) |}_{\partial O} = 0, t > τ . \end{matrix} \end{matrix}

(46)

Notice that system (46) can be viewed as deterministic equation parameterized by

ω \in Ω

. Let assumptions (35)-(39) hold, and then by the Galerkin method similar to [3], we can prove that for any

ω \in Ω

τ \in R

and

u_{τ} \in L^{2} (O)

, Eq. (46) possesses a unique solution

\begin{matrix} u (\cdot, τ, ω, u_{τ}) \in C ([τ, \infty); L^{2} (O)) \cap L_{loc}^{2} ((0, \infty); D_{0}^{1, 2} (O, σ)) \cap L_{loc}^{p} ((0, \infty); L^{p} (O)) . \end{matrix}

(47)

In addition, the solution

u (\cdot, τ, ω, u_{τ})

is continuous in

u_{τ} \in L^{2} (O)

and is

(F, B (L^{2} (O)))

-measurable in

ω \in Ω

. Hence, we can define a continuous cocycle

Ψ : R^{+} \times R \times Ω \times L^{2} (O) \mapsto L^{2} (O)

\begin{matrix} Ψ (t, τ, ω, u_{τ}) = u (t + τ, τ, θ_{- τ} ω, u_{τ}), \forall τ \in R, t > 0, ω \in Ω, u_{τ} \in L^{2} (O) . \end{matrix}

(48)

Let

D_{1} = {D_{1} (τ, ω) : τ \in R, ω \in Ω}

be a family of bounded nonempty subsets of

L^{2} (O)

. A family

D_{1}

is said to be tempered if for any

ν > 0

τ \in R

and

ω \in Ω

, there is

lim_{t \to - \infty} e^{ν t} sup_{u \in D (τ + t, θ_{t} ω)} ∥ u ∥ = 0 .

We denote by

D_{1}

the class of all tempered families of nonempty bounded subsets of

L^{2} (O)

Now, we devote to prove existence of

D_{1}

-pullback random attractors for the cocycle

Ψ

corresponding to Eq. (46) in

L^{2} (O)

Lemma 9.

Suppose (35)-(39) and (17)-(18) hold. Then the continuous cocycle Ψ of problem (46) possesses a closed measurable

D_{1}

-pullback absorbing set

K_{1} = {K_{1} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

, which is given by

\begin{matrix} K_{1} (τ, ω) = {u \in L^{2} {(O) : ∥ u ∥}^{2} \leq R (τ, ω)}, \end{matrix}

(49)

where

\begin{matrix} R (τ, ω) = M_{1} + M_{1} \int_{- \infty}^{0} e^{λ s} {(∥ g (s + τ) ∥}^{2} + | G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s \end{matrix}

(50)

with

M_{1}

is a positive constant independent of

τ, ω

and

D_{1}

Proof.

We first prove that, for any given

τ \in R

and

ω \in Ω

K_{1} (τ, ω)

given by (49) is a pullback absorbing set for the cocycle

Ψ

. Taking inner product of Eq. (46) with u in

L^{2} (O)

, we get

\begin{matrix} \frac{1}{2} \frac{d}{d t} {∥ u ∥}^{2} + {∥ u ∥}_{D_{0}^{1, 2} (O, σ)}^{2} + λ {∥ u ∥}^{2} + \int_{O} f (x, u) u d x \\ = & (g, u) + G_{δ} (θ_{t} ω) \int_{O} h (t, x, u) u d x . \end{matrix}

(51)

By (35) we obtain that

\begin{matrix} \int_{O} f (x, u) u d x \geq α_{1} \int_{O} {| u |}^{p} d x - {∥ β_{1} ∥}_{L^{1} (O)} . \end{matrix}

(52)

By (38) and Young’s inequality, we get

\begin{matrix} G_{δ} (θ_{t} ω) \int_{O} h (t, x, u) u d x \\ \leq & | G_{δ} (θ_{t} ω) | \int_{O} (| ψ_{1} (t, x) {| | u |}^{q} + | ψ_{2} (t, x) | | u |) d x \\ \leq & \frac{α_{1}}{2} \int_{O} {| u |}^{p} d x + C \int_{O} | ψ_{1} (t, x) G_{δ} (θ_{t} ω) |^{\frac{p}{p - q}} d x + C \int_{O} {| ψ_{2} (t, x) G_{δ} (θ_{t} ω) |}^{p_{1}} d x \\ \leq & \frac{α_{1}}{2} \int_{O} {| u |}^{p} d x + C | G_{δ} (θ_{t} ω) |^{\frac{p}{p - q}} ∥ ψ_{1} (t) ∥_{L^{\frac{p}{p - q}}}^{\frac{p}{p - q}} + C | G_{δ} (θ_{t} ω) |^{p_{1}} {∥ ψ_{2} (t) ∥}_{L^{p_{1}}}^{p_{1}} \\ \leq & \frac{α_{1}}{2} \int_{O} {| u |}^{p} d x + C | G_{δ} (θ_{t} ω) |^{\frac{p}{p - q}} + C {| G_{δ} (θ_{t} ω) |}^{p_{1}} . \end{matrix}

(53)

From Cauchy’s inequality, we have

\begin{matrix} (g (t, x), u) \leq \frac{λ}{2} {∥ u ∥}^{2} + \frac{1}{2 λ} {∥ g ∥}^{2} . \end{matrix}

(54)

Therefore, it follows easily from (51)-(54) that

\begin{matrix} \frac{d}{d s} {∥ u ∥}^{2} + {2 ∥ u ∥}_{D_{0}^{1, 2} (O, σ)}^{2} + {λ ∥ u ∥}^{2} + α_{1} {∥ u ∥}_{L^{p} (O)}^{p} \\ \leq & 2 ∥ β_{1} ∥_{L^{1} (O)} + \frac{1}{λ} {∥ g ∥}^{2} + C | G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + C {| G_{δ} (θ_{s} ω) |}^{p_{1}} . \end{matrix}

(55)

Multiplying (55) by

e^{λ s}

, replacing

ω

θ_{- τ} ω

and then integrating with respect to s over

(τ - t, τ)

with

t \geq 0

, we get that

\begin{matrix} ∥ u (τ, τ - t, θ_{- τ} ω, u_{τ - t}) ∥^{2} + 2 \int_{τ - t}^{τ} e^{λ (s - τ)} {∥ u (s, τ - t, θ_{- τ} ω, u_{τ - t}) ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ + α_{1} \int_{τ - t}^{τ} e^{λ (s - τ)} {∥ u ∥}_{L^{p}}^{p} d s \\ \leq & e^{- λ t} ∥ u_{τ - t} ∥^{2} + \frac{2 ∥ β_{1} ∥_{L^{1} (O)}}{λ} + \frac{1}{λ} \int_{τ - t}^{τ} e^{λ (s - τ)} {∥ g (s) ∥}^{2} d s \\ + C \int_{τ - t}^{τ} e^{λ (s - τ)} (| G_{δ} (θ_{s - τ} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s - τ} ω) |^{p_{1}}) d s \\ \leq & e^{- λ t} ∥ u_{τ - t} ∥^{2} + \frac{2 ∥ β_{1} ∥_{L^{1} (O)}}{λ} + \frac{1}{λ} \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ) ∥}^{2} d s \\ + C \int_{- \infty}^{0} e^{λ s} (| G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s . \end{matrix}

(56)

The last two integrals in (56) are well defined due to (17), (42), (44) and the continuity of

ω

. For every

u_{τ - t} \in D_{1} (τ - t, θ_{- t} ω)

and

D_{1} = {D_{1} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

, we have

\begin{matrix} \underset{t \to + \infty}{lim sup} e^{- λ t} {∥ u_{τ - t} ∥}^{2} = 0 . \end{matrix}

(57)

Hence, there exists some

T_{1} = T_{1} (σ, τ, ω, D_{1}) > 0

such that for all

t \geq T_{1}

\begin{matrix} ∥ u (τ, τ - t, θ_{- τ} ω, u_{τ - t}) ∥^{2} + 2 \int_{τ - t}^{τ} e^{λ (s - τ)} {∥ u (s, τ - t, θ_{- τ} ω, u_{τ - t}) ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ + α_{1} \int_{τ - t}^{τ} e^{λ (s - τ)} {∥ u ∥}_{L^{p}}^{p} d s \\ \leq & 1 + \frac{2 ∥ β_{1} ∥_{L^{1} (O)}}{λ} + \frac{1}{λ} \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ) ∥}^{2} d s + C \int_{- \infty}^{0} e^{λ s} (| G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s \\ \leq & M_{1} + M_{1} \int_{- \infty}^{0} e^{λ s} {(∥ g (s + τ) ∥}^{2} + | G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s, \end{matrix}

(58)

where

M_{1}

is a positive constant independent of

τ, ω

and

D_{1}

. Then by (58) we get that, for every

τ \in R

ω \in Ω

and every

D_{1} \in D_{1}

K_{1} (τ, ω)

given by (49) satisfies

Ψ (t, τ - t, θ_{- t} ω, D_{1} (τ - t, θ_{- t} ω)) \subseteq K_{1} (τ, ω) .

We next prove that

K_{1} \in D_{1}

. Let

ν

be an arbitrary positive constant. Then for each

τ \in R

and

ω \in Ω

, we can get from (50) that

\begin{matrix} e^{ν t} {∥ K_{1} (τ + t, θ_{t} ω) ∥}^{2} \leq e^{ν t} R (τ + t, θ_{t} ω) \\ = & M_{1} e^{ν t} + M_{1} e^{ν t} \int_{- \infty}^{0} e^{λ s} ({∥ g (s + t + τ) ∥}^{2} + | G_{δ} (θ_{s + t} ω) |^{\frac{p}{p - q}} + {| G_{δ} (θ_{s + t} ω) |}^{p_{1}}) d s . \end{matrix}

(59)

First, we can get from (18) that

\begin{matrix} lim_{t \to - \infty} e^{ν t} \int_{- \infty}^{0} e^{λ s} {∥ g (s + τ + t) ∥}^{2} d s \\ = & lim_{t \to - \infty} e^{- λ τ} e^{ν t} \int_{- \infty}^{τ} e^{λ s} {∥ g (s + t) ∥}^{2} d s = 0 . \end{matrix}

(60)

Let

\tilde{ν} = min {λ, ν}

, and then we can get from (42) and (44) that for any

t \leq 0

\begin{matrix} e^{ν t} \int_{- \infty}^{0} e^{λ s} (| G_{δ} (θ_{s + t} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s + t} ω) |^{p_{1}}) d s \\ \leq & \int_{- \infty}^{t} e^{\tilde{ν} s} (| G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s . \end{matrix}

(61)

Note that

\int_{- \infty}^{0} e^{\tilde{ν} s} (| G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s < + \infty,

which implies that

\begin{matrix} \int_{- \infty}^{t} e^{\tilde{ν} s} (| G_{δ} (θ_{s} ω) |^{\frac{p}{p - q}} + | G_{δ} (θ_{s} ω) |^{p_{1}}) d s \to 0 a s t \to - \infty . \end{matrix}

(62)

It follows from (59)-(62) that

K_{1}

is tempered, i.e.

K_{1} \in D_{1}

. Moreover, since for each

τ \in R

R (τ, \cdot) : Ω \mapsto R

(F, B (R))

-measurable, then

K_{1} (τ, \cdot)

is also measurable. Hence,

K_{1} \in D_{1}

is a closed measurable

D_{1}

-pullback absorbing set for

Ψ

. The proof is completed. □

Lemma 10.

Let (35)-(39) hold. Then for each

τ \in R

t > τ

ω \in Ω

and for each bounded sequence

{u_{0, n}}_{n = 1}^{\infty} \subseteq L^{2} (O)

, the sequence

{u (t, τ, ω, u_{0, n})}_{n = 1}^{\infty}

possesses a convergent subsequence in

L^{2} (O)

Proof.

Taking

T > t

, and integrating (55) over

[τ, T]

, we can get that

\begin{matrix} {u (\cdot, τ, ω, u_{0, n})}_{n = 1}^{\infty} is bounded in L^{p} ((τ, T); L^{p} (O)) \cap L^{2} ((τ, T); D_{0}^{1, 2} (O, σ)) . \end{matrix}

(63)

We can also infer from (36), (38) and (63) that, for

s \in [τ, T]

\begin{matrix} {f (\cdot, u (\cdot, τ, ω, u_{0, n}))}_{n = 1}^{\infty} & and {h (\cdot, \cdot, u (\cdot, τ, ω, u_{0, n})) G_{δ} (θ_{s} ω)}_{n = 1}^{\infty} \\ are bounded in & L^{p_{1}} ((τ, T); L^{p_{1}} (O)) . \end{matrix}

(64)

Then it follows from (63), (64), and Eq. (46), that

\begin{matrix} {\frac{\partial}{\partial t} u (\cdot, τ, ω, u_{0, n})}_{n = 1}^{\infty} is bounded in L^{2} ((τ, T); D_{0}^{- 1, 2} (O, σ)) + L^{p_{1}} ((τ, T); L^{p_{1}} (O)) . \end{matrix}

(65)

By Lemma 3, we note that the embedding

D_{0}^{1, 2} (O, σ) ↪ L^{2} (O)

is compact (in both cases of bounded and unbounded domain). Then we can get from (63), (65) and Aubin-Lions compactness lemma that there exist some

w \in L^{2} ((τ, T); L^{2} (O))

and a subsequence of

{u (s, τ, ω, u_{0, n})}_{n = 1}^{\infty}

such that

\begin{matrix} u (\cdot, τ, ω, u_{0, n_{k}}) \to w in L^{2} ((τ, T); L^{2} (O)) . \end{matrix}

(66)

By choosing a further subsequence (relabelled the same), we infer from (66) that

\begin{matrix} u (s, τ, ω, u_{0, n_{k}}) \to w (s) in L^{2} (O), a . e . s \in [τ, T] . \end{matrix}

(67)

Finally, since

t \in (τ, T)

, we can by the continuity of solutions on initial data in

L^{2} (O)

and (67) get

\begin{matrix} u (t, τ, ω, u_{0, n_{k}}) = u (t, s, ω, u (s, τ, ω, u_{0, n_{k}})) \to u (t, s, ω, w (s)), \end{matrix}

i.e.,

u (t, τ, ω, u_{0, n})

possesses a convergent subsequence in

L^{2} (O)

. We complete the proof. □

Lemma 11.

Suppose (35)-(39) and (17) hold. Then the continuous cocycle Ψ for Eq. (46) is

D_{1}

-pullback asymptotically compact in

L^{2} (O)

Proof.

For any

τ \in R

ω \in Ω

D_{1} = {D_{1} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

t_{n} \to + \infty

n \to \infty

and

u_{0, n} \in D_{1} (τ - t_{n}, θ_{- t_{n}} ω)

, we shall prove the sequence

Ψ (t_{n}, τ - t_{n}, θ_{- t_{n}} ω, u_{0, n})

has a convergent subsequence in

L^{2} (O)

. Note that

t_{n} \to + \infty

n \to \infty

and

u_{0, n} \in D_{1} (τ - t_{n}, θ_{- t_{n}} ω)

. We can get from Lemma 9 that there exist

N_{1} = N_{1} (τ, ω, D_{1}) > 0

such that for all

n \geq N_{1}

that

\begin{matrix} ∥ u (τ, τ - t_{n}, θ_{- τ} ω, u_{0, n}) ∥ \leq C (τ, ω), \end{matrix}

(68)

which implies that

\begin{matrix} {u (τ, τ - t_{n}, θ_{- τ} ω, u_{0, n})}_{n = 1}^{\infty} is bounded in L^{2} (O) . \end{matrix}

(69)

It follows from (69) and Lemma 10 that the sequence

\begin{matrix} {u (τ, τ - t_{n}, θ_{- τ} ω, u_{0, n})}_{n = 1}^{\infty} is precompact in L^{2} (O), \end{matrix}

which along with

Ψ (t_{n}, τ - t_{n}, θ_{- t_{n}} ω, u_{0, n}) = u (τ, τ - t_{n}, θ_{- τ} ω, u_{0, n})

, it implies the result. □

Theorem 2.

Suppose (35)-(39) and (17)-(18) hold. Then the continuous cocycle Ψ associated with system (46) possesses a unique

D_{1}

-pullback random attractor

A = {A (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

L^{2} (O)

Proof.

From Lemma 9, Lemma 11 as well as [34], the existence of unique

D_{1}

-pullback random attractor

A = {A (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

follows.

□

4.2. Stochastic Semi-linear Degenerate Parabolic Equation Driven by linear Multiplicative Noise

In this subsection, we discuss the following stochastic semi-linear degenerate parabolic equation:

\begin{matrix} \{\begin{matrix} \frac{\partial u}{\partial t} - div (σ (x) \nabla u) + λ u + f (x, u) = g (t, x) + u \circ \frac{d w}{d t}, t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (x, t) |}_{\partial O} = 0, t > τ, \end{matrix} \end{matrix}

(70)

and consider the following Wong-Zakai approximate system for Eq. (70):

\begin{matrix} \{\begin{matrix} \frac{\partial u_{δ}}{\partial t} - div (σ (x) \nabla u_{δ}) + λ u_{δ} + f (x, u_{δ}) = g (t, x) + u_{δ} G_{δ} (θ_{t} ω), t > τ, \\ u_{δ} (τ, x) = u_{δ, τ} (x), τ \in R, \\ u_{δ} {(x, t) |}_{\partial O} = 0, t > τ . \end{matrix} \end{matrix}

(71)

We will investigate the relations between the solutions of Eq. (70) and Eq. (71). To this end, we need to transform the stochastic equation (70) into a pathwise deterministic one. Let

\begin{matrix} v (t, τ, ω) = e^{- ω (t)} u (t, τ, ω), \end{matrix}

(72)

with

v_{τ} = e^{- ω (τ)} u_{τ} .

Then by (70) and (72), we get

\begin{matrix} \{\begin{matrix} \frac{\partial v}{\partial t} + A v + λ v + e^{- ω (t)} f (x, u) = e^{- ω (t)} g (t, x), t > τ, \\ u (τ, x) = u_{τ} (x), τ \in R, \\ {u (x, t) |}_{\partial O} = 0, t > τ, \end{matrix} \end{matrix}

(73)

where

A v = - div (σ (x) \nabla v)

. We also introduce a similar transform for Eq. (71) as we did for Eq. (70). Let

\begin{matrix} v_{δ} (t, τ, ω, v_{δ, τ}) = e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} u_{δ} (t, τ, ω, u_{δ, τ}) \end{matrix}

(74)

with

v_{δ, τ} = e^{- \int_{0}^{τ} G_{δ} (θ_{r} ω) d r} u_{δ, τ} .

Then we have

\begin{matrix} \{\begin{matrix} \frac{\partial v_{δ}}{\partial t} + A v_{δ} + λ v_{δ} + e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ}) = e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} g (t, x), t > τ, \\ v (τ, x) = v_{δ, τ} (x), τ \in R, \\ {v (x, t) |}_{\partial O} = 0, t > τ . \end{matrix} \end{matrix}

(75)

For any

ω \in Ω, τ \in R

and

v_{τ} \in L^{2} (O)

, let (35)-(37) hold. Then by the classic Galerkin method, we can get the existence and uniqueness of solution

v (\cdot, τ, ω,

υ_{τ}) \in C ([τ, \infty), L^{2} (O))

for system (73). In addition

v (\cdot, τ, ω, υ_{τ})

is continuous in

v_{τ} \in L^{2} (O)

and is

(F, B (L^{2} (O)))

-measurable in

ω \in Ω

. Thus, we can define a continuous cocycle

{\tilde{Ψ}}_{0} : R^{+} \times R \times Ω \times L^{2} (O) \mapsto L^{2} (O)

for system (70) by

\begin{matrix} {\tilde{Ψ}}_{0} (t, τ, ω, u_{τ}) = u (t + τ, τ, θ_{- τ} ω, u_{τ}) = e^{ω (t) - ω (- τ)} v (t + τ, τ, θ_{- τ} ω, v_{τ}) . \end{matrix}

(76)

Similarly, we can also define a continuous cocycle

{\tilde{Ψ}}_{δ} (t, τ, ω, u_{δ, τ})

for system (71).

Lemma 12.

Assume (35)-(37) and (17)-(18) hold. Then the continuous cocycle

{\tilde{Ψ}}_{0}

for system (70) possesses a closed measurable

D_{1}

-pullback absorbing set

{\tilde{B}}_{0} = {{\tilde{B}}_{0} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

, which is given by

\begin{matrix} {\tilde{B}}_{0} (τ, ω) = {u \in L^{2} {(O) : ∥ u ∥}^{2} \leq {\tilde{R}}_{0} (τ, ω)}, \end{matrix}

(77)

where

\begin{matrix} {\tilde{R}}_{0} (τ, ω) = 4 \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(78)

Proof.

Taking inner product of Eq. (73) with

v (t, τ, ω) = e^{- ω (t)} u (t, τ, ω)

, we have

\begin{matrix} \frac{1}{2} \frac{d}{d t} {∥ v ∥}^{2} + {∥ v ∥}_{D_{0}^{1, 2} (O, σ)}^{2} + λ {∥ v ∥}^{2} + e^{- ω (t)} \int_{O} f (x, u) v d x = e^{- ω (t)} (g, v) . \end{matrix}

(79)

It follows from (35) and (72) that

\begin{matrix} e^{- ω (t)} \int_{O} f (x, u) v d x \geq α_{1} e^{- 2 ω (t)} {∥ u ∥}_{L^{p} (O)}^{p} - {∥ β_{1} ∥}_{L^{1} (O)} e^{- 2 ω (t)} . \end{matrix}

(80)

By Cauchy’s inequality, we obtain

\begin{matrix} e^{- ω (t)} (g, v) \leq \frac{λ}{4} {∥ v ∥}^{2} + \frac{1}{λ} e^{- 2 ω (t)} {∥ g ∥}^{2} . \end{matrix}

(81)

Then combining (79)-(81), we have

\begin{matrix} \frac{d}{d s} {∥ v ∥}^{2} & + {2 ∥ v ∥}_{D_{0}^{1, 2} (O, σ)}^{2} + \frac{3}{2} {λ ∥ v ∥}^{2} + 2 α_{1} e^{- 2 ω (s)} {∥ u ∥}_{L^{p} (O)}^{p} \\ \leq & 2 ∥ β_{1} ∥_{L^{1} (O)} e^{- 2 ω (s)} + \frac{2}{λ} e^{- 2 ω (s)} {∥ g ∥}^{2} . \end{matrix}

(82)

Multiplying

e^{\frac{3}{2} λ s}

on both sides of (82) and then integrating with respect to s over

[τ - t, τ]

with

t > 0

, we get

\begin{matrix} ∥ v (τ, & τ - t, ω, v_{τ - t} ∥^{2} + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} {∥ v ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s + 2 α_{1} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 ω (s)} {∥ u ∥}_{L^{p} (O)}^{p} d s \\ \leq & 2 ∥ β_{1} ∥_{L^{1} (O)} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 ω (s)} d s + \frac{2}{λ} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 ω (s)} {∥ g (s) ∥}^{2} d s + {∥ v_{τ - t} ∥}^{2} e^{- \frac{3}{2} λ t} . \end{matrix}

(83)

Replacing

ω

in (83) by

θ_{- τ} ω

and using

\begin{matrix} u (s, τ - t, θ_{- τ} ω, u_{τ - t}) = e^{(ω (- τ + s) - ω (- τ))} v (s, τ - t, θ_{- τ} ω, u_{τ - t}), \end{matrix}

(84)

we get that

\begin{matrix} ∥ u ( & τ, τ - t, θ_{- τ} ω, u_{τ - t} {) ∥}^{2} e^{2 ω (- τ)} + 2 α_{1} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 (ω (- τ + s) - ω (- τ))} {∥ u ∥}_{L^{p} (O)}^{p} d s \\ + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{- 2 ω (s - τ) + 2 ω (- τ)} {∥ u ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ \leq & 2 ∥ β_{1} ∥_{L^{1} (O)} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 (ω (- τ + s) - ω (- τ))} d s + e^{2 ω (- τ) - 2 ω (- t)} {∥ u_{τ - t} ∥}^{2} e^{- \frac{3}{2} λ t} \\ + \frac{2}{λ} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ) - 2 (ω (- τ + s) - ω (- τ))} {∥ g (s) ∥}^{2} d s . \end{matrix}

(85)

Then from (85), we get

\begin{matrix} ∥ u ( & τ, τ - t, θ_{- τ} ω, u_{τ - t} {) ∥}^{2} + 2 α_{1} \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} {∥ u ∥}_{L^{p} (O)}^{p} d s \\ + 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s} e^{- 2 ω (s)} {∥ u ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ \leq & 2 ∥ β_{1} ∥_{L^{1} (O)} \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} d s + \frac{2}{λ} \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} {∥ g (s + τ) ∥}^{2} d s \\ + e^{- 2 ω (- t)} e^{- \frac{3}{2} λ t} {∥ u_{τ - t} ∥}^{2} . \end{matrix}

(86)

By (17) and (42), we have

\begin{matrix} 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s < \infty . \end{matrix}

(87)

Note that if

u_{τ - t} \in D_{1} (τ - t, θ_{- t} ω)

and

D_{1} \in D_{1}

, then by (42) we have

\begin{matrix} \underset{t \to + \infty}{lim sup} e^{- 2 ω (- t)} e^{- \frac{3}{2} λ t} {∥ u_{τ - t} ∥}^{2} = 0 . \end{matrix}

(88)

Then there exists some

T_{4} = T_{4} (τ, ω, D_{1}) > 0

such that for all

t \geq T_{4}

\begin{matrix} e^{- 2 ω (- t)} e^{- \frac{3}{2} λ t} ∥ u_{τ - t} ∥^{2} \leq 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s - 2 ω (s)} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s, \end{matrix}

(89)

which along with (76) implies that

\begin{matrix} {\tilde{Ψ}}_{0} (t, τ - t, θ_{- t} ω, D (τ - t, θ_{- t} ω)) \subseteq {\tilde{B}}_{0} (τ, ω), \forall t \geq T_{4}, \end{matrix}

where

{\tilde{B}}_{0} (τ, ω)

is given by (77). In addition, by (18), (42) and the continuity of

ω (t)

, we can easily get that

{\tilde{B}}_{0}

is tempered, that is,

{\tilde{B}}_{0} \in D_{1}

. Hence,

{\tilde{B}}_{0} \in D_{1}

is a closed measurable

D_{1}

-pullback absorbing set for

{\tilde{Ψ}}_{0}

. The proof is completed. □

Theorem 3.

Suppose (35)-(37) and (17)-(18) hold. Then the continuous cocycle

{\tilde{Ψ}}_{0}

for system (70) is

D_{1}

-pullback asymptotically compact and possesses a unique

D_{1}

-pullback random attractor

{\tilde{A}}_{0} = {{\tilde{A}}_{0} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

L^{2} (O)

Proof.

The proof of

D_{1}

-pullback asymptotical compactness of cocycle

Ψ_{0}

L^{2} (O)

is similar to that of Lemma 11. And then by [34] Proposition 2.1 and Lemma 12, we can easily get the cocycle

Ψ_{0}

possesses a unique

D_{1}

-pullback random attractor

A_{0}

. □

Lemma 13.

Suppose (35)-(37) and (17)-(18) hold. Then the continuous cocycle

{\tilde{Ψ}}_{δ}

for Eq. (71) possesses a closed measurable

D_{1}

-pullback absorbing set

{\tilde{B}}_{δ} = {{\tilde{B}}_{δ} (τ, ω) : τ \in R, ω \in Ω} \in D_{1},

\begin{matrix} {\tilde{B}}_{δ} (τ, ω) = {u_{δ} \in L^{2} (O) : ∥ u_{δ} ∥^{2} \leq {\tilde{R}}_{δ} (τ, ω)}, \end{matrix}

(90)

where

\begin{matrix} {\tilde{R}}_{δ} (τ, ω) = 4 \int_{- \infty}^{0} e^{\frac{3}{2} λ s} e^{2 \int_{s}^{0} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(91)

In addition, we have for every

τ \in R

and

ω \in Ω

\begin{matrix} lim_{δ \to 0} {\tilde{R}}_{δ} (τ, ω) = {\tilde{R}}_{0} (τ, ω), \end{matrix}

(92)

where

{\tilde{R}}_{0} (τ, ω)

is given by (78).

Proof.

By (75) we get

\begin{matrix} \frac{1}{2} \frac{d}{d t} ∥ v_{δ} ∥^{2} + {∥ v_{δ} ∥}_{D_{0}^{1, 2} (O, σ)}^{2} & + λ ∥ v_{δ} ∥^{2} + \int_{O} e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ}) v_{δ} d x \\ = e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} (g, v_{δ}) . \end{matrix}

(93)

By (35) and (74), we get

\begin{matrix} - \int_{O} e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ}) v_{δ} d x \\ \leq & - α_{1} e^{- 2 \int_{0}^{t} G_{δ} (θ_{r} ω) d r} ∥ u_{δ} ∥_{L^{p} (O)}^{p} + {∥ β_{1} ∥}_{L^{1} (O)} e^{- 2 \int_{0}^{t} G_{δ} (θ_{r} ω) d r} . \end{matrix}

(94)

By Cauchy’s inequality, we obtain

\begin{matrix} e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} (g, v_{δ}) \leq \frac{λ}{4} ∥ v_{δ} ∥^{2} + \frac{1}{λ} e^{- 2 \int_{0}^{t} G_{δ} (θ_{r} ω) d r} {∥ g ∥}^{2} . \end{matrix}

(95)

Then it follows from (93)-(95) that

\begin{matrix} \frac{d}{d s} ∥ v_{δ} ∥^{2} + 2 {∥ v_{δ} ∥}_{D_{0}^{1, 2} (O, σ)}^{2} & + \frac{3}{2} λ ∥ v_{δ} ∥^{2} + 2 α_{1} e^{- 2 \int_{0}^{s} G_{δ} (θ_{r} ω) d r} {∥ u_{δ} ∥}_{L^{p} (O)}^{p} \\ \leq 2 e^{- 2 \int_{0}^{s} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) . \end{matrix}

(96)

For all

τ \in R, t \in R^{+}

and

ω \in Ω

, multiplying

e^{\frac{3}{2} λ s}

and then integrating with respect to s from

τ - t

τ

, we have

\begin{matrix} ∥ v_{δ} & (τ, τ - t, ω, v_{δ, τ - t}) ∥^{2} + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} {∥ v_{δ} ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ + 2 α_{1} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{- 2 \int_{0}^{s} G_{δ} (θ_{r} ω) d r} {∥ u_{δ} ∥}_{L^{p} (O)}^{p} d s \\ \leq & e^{- \frac{3}{2} λ t} ∥ v_{δ, τ - t} ∥^{2} + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{- 2 \int_{0}^{s} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g (s) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(97)

Replacing

ω

in (97) by

θ_{- τ} ω

, we get

\begin{matrix} ∥ v_{δ} (τ, & τ - t, θ_{- τ} ω, v_{δ, τ - t} {) ∥}^{2} + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} {∥ v_{δ} ∥}_{D_{0}^{1, 2} (O, σ)}^{2} d s \\ + 2 α_{1} \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{- 2 \int_{0}^{s} G_{δ} (θ_{r - τ} ω) d r} {∥ u_{δ} ∥}_{L^{p} (O)}^{p} d s \\ \leq & e^{- \frac{3}{2} λ t} ∥ v_{δ, τ - t} ∥^{2} + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{- 2 \int_{0}^{s} G_{δ} (θ_{r - τ} ω) d r} (\frac{1}{λ} {∥ g (s) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(98)

By (74) and (98) we get

\begin{matrix} ∥ u_{δ} (τ, τ - t, θ_{- τ} ω, u_{δ, τ - t}) ∥^{2} \\ \leq & e^{- \frac{3}{2} λ t} e^{2 \int_{τ - t}^{τ} G_{δ} (θ_{r - τ} ω) d r} {∥ u_{δ, τ - t} ∥}^{2} \\ + 2 \int_{τ - t}^{τ} e^{\frac{3}{2} λ (s - τ)} e^{2 \int_{s}^{τ} G_{δ} (θ_{r - τ} ω) d r} (\frac{1}{λ} {∥ g (s) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s \\ \leq & e^{- \frac{3}{2} λ t} e^{2 \int_{- t}^{0} G_{δ} (θ_{r} ω) d r} {∥ u_{δ, τ - t} ∥}^{2} \\ + 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s + 2 \int_{s}^{0} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(99)

By (17), (42), (44) and the continuity of

ω (t)

we get

\begin{matrix} 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s + 2 \int_{s}^{0} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s < \infty . \end{matrix}

(100)

Note that if

u_{δ, τ - t} \in D_{1} (τ - t, θ_{- t} ω)

and

D_{1} \in D_{1}

, then by (42), (44) and the continuity of

ω (t)

, we get

\begin{matrix} \underset{t \to + \infty}{lim sup} e^{- \frac{3}{2} λ t} e^{2 \int_{- t}^{0} G_{δ} (θ_{r} ω) d r} {∥ u_{δ, τ - t} ∥}^{2} = 0, \end{matrix}

(101)

which implies that there exists

T_{5} = T_{5} (τ, ω, D_{1}, δ) > 0

such that for all

t \geq T_{5}

\begin{matrix} e^{- \frac{3}{2} λ t} e^{2 \int_{- t}^{0} G_{δ} (θ_{r} ω) d r} {∥ u_{δ, τ - t} ∥}^{2} \\ \leq & 2 \int_{- \infty}^{0} e^{\frac{3}{2} λ s + 2 \int_{s}^{0} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} {∥ g (s + τ) ∥}^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(102)

By (99)-(102), we get

\begin{matrix} ∥ u_{δ} (τ, τ - t, θ_{- τ} ω, u_{δ, τ - t}) ∥^{2} \leq 4 \int_{- \infty}^{0} e^{\frac{3}{2} λ s + 2 \int_{s}^{0} G_{δ} (θ_{r} ω) d r} (\frac{1}{λ} ∥ g (s + τ) ∥^{2} + ∥ β_{1} ∥_{L^{1} (O)}) d s . \end{matrix}

(103)

In other words, we get for all

t \geq T_{5}

\begin{matrix} u_{δ} (τ, τ - t, θ_{- τ} ω, D (τ - t, θ_{- t} ω)) \subseteq {\tilde{B}}_{δ} (τ, ω), \end{matrix}

(104)

where

{\tilde{B}}_{δ} (τ, ω)

is given by (90). In addition,

{\tilde{B}}_{δ}

is tempered due to (18), (42), (44). Therefore,

{\tilde{B}}_{δ}

is a closed measurable

D_{1}

-pullback absorbing set of

Ψ_{δ}

. The proof of (92) is similar to that of [21] Lemma 3.7 and the details are omitted here. □

Theorem 4.

Suppose (35)-(37) and (17)-(18) hold. Then the cotinuous cocycle

{\tilde{Ψ}}_{δ}

for Eq. (71) is

D_{1}

-pullback asymptotically compact and possesses a unique

D_{1}

-pullback random attractor

{\tilde{A}}_{δ} = {{\tilde{A}}_{δ} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

L^{2} (O)

Proof.

The proof is similar to that of Theorem 3. □

Now, we show the solution of Eq. (71) converges to the solution of Eq. (70) as

δ \to 0

. Toward this end, we further assume the following assumption hold: there exists some

α_{4} > 0

such that for all

x \in O

u \in R

\begin{matrix} | \frac{\partial f}{\partial u} (x, u) | \leq α_{4} {(1 + | u |}^{p - 2}) . \end{matrix}

(105)

Lemma 14.

Suppose (35)-(37) and (17)-(18) hold. Let u and

u_{δ}

be the solutions of Eq. (70) and Eq. (71), respectively, with initial data

u_{τ}

and

u_{δ, τ}

. If

u_{δ, τ} \to u_{τ}

L^{2} (O)

δ \to 0

, then for every

τ \in R, ω \in Ω

and

T > 0

, there exists some

{\tilde{δ}}_{0} = {\tilde{δ}}_{0} (τ, ω, T) > 0

such that for any

0 < | δ | < {\tilde{δ}}_{0}

and

t \in [τ, τ + T]

u_{δ} (t, τ, ω, u_{δ, τ}) \to u (t, τ, ω, u_{τ})

L^{2} (O)

Proof.

Let

ξ = v_{δ} - v

and then we have

\begin{matrix} \frac{1}{2} \frac{d}{d t} {∥ ξ ∥}^{2} + {∥ ξ ∥}_{D_{0}^{1, 2} (O, σ)}^{2} + λ {∥ ξ ∥}^{2} \\ = & \int_{O} (e^{- ω (t)} f (x, u) - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ})) ξ d x + (e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} - e^{- ω (t)}) (g (t), ξ) . \end{matrix}

(106)

By using (36)-(37) and (105), we have

\begin{matrix} (e^{- ω (t)} f (x, u) - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ})) ξ \\ = & (e^{- ω (t)} f (x, e^{ω (t)} v) - e^{- ω (t)} f (x, v_{δ} e^{ω (t)})) ξ + (e^{- ω (t)} f (x, v_{δ} e^{ω (t)}) - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, v_{δ} e^{ω (t)})) ξ \\ + (e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, v_{δ} e^{ω (t)}) - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, v_{δ} e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r})) ξ \\ = & \frac{\partial f}{\partial s} (e^{- ω (t)} (v e^{ω (t)} - v_{δ} e^{ω (t)})) ξ + f (x, v_{δ} e^{ω (t)}) (e^{- ω (t)} - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r}) ξ \\ + e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} v_{δ} (e^{ω (t)} - e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r}) \frac{\partial f}{\partial s} ξ \\ = & - \frac{\partial f}{\partial s} ξ^{2} + f (x, v_{δ} e^{ω (t)}) ξ (e^{- ω (t)} - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r}) + v_{δ} (e^{ω (t) - \int_{0}^{t} G_{δ} (θ_{r} ω) d r} - 1) \frac{\partial f}{\partial s} ξ \\ \leq & ∣ β_{3} ∣ {| ξ |}^{2} + (α_{2} e^{(p - 1) ω (t)} | v_{δ} |^{p - 1} | ξ | + | β_{2} | | ξ |) | e^{- ω (t)} - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} | \\ + α_{4} | 1 - e^{ω (t) - \int_{0}^{t} G_{δ} (θ_{r} ω) d r} | (| v_{δ} |^{p - 1} | e^{ω (t)} + e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r} |^{p - 2} | ξ | + | v_{δ} | | ξ |) . \end{matrix}

(107)

From Lemma 8, we find that for any

ϵ > 0

, there exists some

{\tilde{δ}}_{1} = {\tilde{δ}}_{1} (ϵ, τ, ω, T) > 0

such that

\begin{matrix} | 1 - e^{ω (t) - \int_{0}^{t} G_{δ} (θ_{r} ω) d r} | < ϵ, | e^{- ω (t)} - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} | < ϵ, \forall 0 < | δ | < {\tilde{δ}}_{1}, t \in [τ, τ + T] . \end{matrix}

(108)

It follows from (107) and (108) that

\begin{matrix} \int_{O} (e^{- ω (t)} f (x, u) - e^{- \int_{0}^{t} G_{δ} (θ_{r} ω) d r} f (x, u_{δ})) ξ d x \leq {C ∥ ξ ∥}^{2} + C ϵ (∥ v_{δ} ∥_{L^{p} (O)}^{p} + {∥ v ∥}_{L^{p} (O)}^{p} + 1) . \end{matrix}

(109)

By Cauchy’s inequality, we have

\begin{matrix} (e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r} - e^{- ω (t)}) (g (t), ξ) \leq | e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r} - e^{- ω (t)} | (\frac{1}{2} {∥ g ∥}^{2} + \frac{1}{2} {∥ ξ ∥}^{2}) . \end{matrix}

(110)

Combing (106)-(110), we get

\begin{matrix} \frac{d}{d t} {∥ ξ ∥}^{2} \leq {C ∥ ξ ∥}^{2} + C ϵ (∥ v_{δ} ∥_{L^{p} (O)}^{p} + {∥ v ∥}_{L^{p} (O)}^{p} + {∥ g ∥}^{2} + 1) . \end{matrix}

(111)

Applying Gronwall’s inequality to (111), we get for all

0 < | δ | < {\tilde{δ}}_{1}

and

t \in [τ, τ + T],

\begin{matrix} {∥ ξ (t) ∥}^{2} \leq & e^{C (t - τ)} {∥ ξ (τ) ∥}^{2} + C ϵ e^{C (t - τ)} \int_{τ}^{t} (1 + {∥ v_{δ} (s, τ, ω, v_{δ, τ}) ∥}_{L^{p} (O)}^{p} \\ + ∥ v (s, τ, ω, v_{τ}) ∥_{L^{p} (O)}^{p} + {∥ g (s) ∥}^{2}) d s . \end{matrix}

(112)

By (72), (74), (83), (97) and (112), we get that there exists some

{\tilde{δ}}_{2} \in (0, {\tilde{δ}}_{1})

and

\tilde{c_{1}} = {\tilde{c}}_{1} (τ, T, ω) > 0

such that for all

0 < | δ | < {\tilde{δ}}_{2}

and

t \in [τ, τ + T]

\begin{matrix} ∥ v_{δ} (t, τ, ω, v_{δ, τ}) - v (t, τ, ω, v_{τ}) ∥^{2} \\ \leq & e^{{\tilde{c}}_{1} (t - τ)} {∥ v_{δ, τ} - v_{τ} ∥}^{2} + {\tilde{c}}_{1} ϵ e^{{\tilde{c}}_{1} (t - τ)} (1 + ∥ v_{τ} ∥^{2} + ∥ v_{δ, τ} ∥^{2} + \int_{τ}^{t} {∥ g (s) ∥}^{2} d s) . \end{matrix}

(113)

Using (72) and (74) again, we get

\begin{matrix} ∥ u_{δ} (t, τ, ω, u_{δ, τ}) - u (t, τ, ω, u_{τ}) ∥ \\ \leq & ∥ v_{δ} (t, τ, ω, v_{δ, τ}) - v (t, τ, ω, v_{τ}) ∥ | e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r} | + | e^{\int_{0}^{t} G_{δ} (θ_{r} ω) d r} - e^{ω (t)} | ∥ v (t, τ, ω, v_{τ}) ∥ . \end{matrix}

(114)

Note that

u_{δ, τ} = v_{δ, τ} e^{\int_{0}^{τ} G_{δ} (θ_{r} ω) d r}

and

u_{τ} = v_{τ} e^{ω (τ)}

. Then by the continuity of

ω (t)

, (45), (83), and (113)-(114), we can obtain the desired convergence. □

Lemma 15.

Suppose (35)-(37) and (17)-(18) hold. For any given

τ \in R, T > 0

and

ω \in Ω

, if

δ_{n} \to 0

and

u_{n} \in {\tilde{A}}_{δ_{n}} (τ, ω),

then the sequence

{u_{n}}_{n = 1}^{\infty}

has a convergent subsequence in

L^{2} (O)

Proof.

By using Lemma 3 and the similar method as that of Lemma 3.10 in [21], we can get the result. □

Theorem 5.

Suppose (35)-(36) and (17)-(18) hold. Then for any given

τ \in R

and

ω \in Ω

, the following relationship holds:

\begin{matrix} lim_{δ \to 0} d_{L^{2} (O)} ({\tilde{A}}_{δ} (τ, ω), {\tilde{A}}_{0} (τ, ω)) = 0 . \end{matrix}

Proof.

By Lemma 12 and Lemma 13, we obtain that, for any

τ \in R

and

ω \in Ω

\begin{matrix} lim_{δ \to 0} ∥ {\tilde{B}}_{δ} {(τ, ω) ∥}^{2} = {∥ {\tilde{B}}_{0} (τ, ω) ∥}^{2} \leq {\tilde{B}}_{0} (τ, ω), \end{matrix}

where

{\tilde{B}}_{0} (τ, ω)

is given by (78) and

{\tilde{B}}_{0} = {{\tilde{B}}_{0} (τ, ω) : τ \in R, ω \in Ω} \in D_{1}

. Let

δ \to 0

and

u_{δ} \to u_{τ}

, and then from Lemma 14, we get, for every

τ \in R

t \in R^{+}

, and

ω \in Ω

, that

Ψ_{δ} (τ, t, ω, u_{δ, τ}) \to Ψ_{0} (τ, t, ω, u_{τ})

L^{2} (O)

. Then, by Lemma 15 and Theorem 3.1 in [28] we can get the result. □

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Dynamics of Non-autonomous Stochastic Semi-linear Degenerate Parabolic Equations with Nonlinear Noise

Abstract

1. Introduction

2. Preliminaries

2.1. Functional setting

2.2. Theory of Random Attractors

3. Mean Random Attractors for Stochastic Semi-linear Degenerate Parabolic Equation

4. Wong-ZaKai Approximations of Stochastic Semi-linear Degenerate Parabolic Equation

4.1. Random dynamical systems for Wong-Zakai Approximations

4.2. Stochastic Semi-linear Degenerate Parabolic Equation Driven by linear Multiplicative Noise

References

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