Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on ${\mathbb{R}}^N$

1.   Introduction

In this paper, we consider the regularity of Wong-Zakai approximations of a stochastic quasi-linear parabolic equation (with $p$-Laplacian) with multiplicative white noise on the entire space ${\mathbb{R}}^N, 1\leq N\in {\mathbb{N}}$: For $t>\tau$ and $x\in {\mathbb{R}}^N$,

where $g\in L_{loc}^2( {\mathbb{R}}, L^2( {\mathbb{R}}^N))$, $\lambda>0$ and $p>2$ are constants, $f$ is a nonlinear function satisfying some dissipative conditions, $W(t)$ is two-sided real-valued Brownian motions on a probability space, $\circ$ signifies the Stratonovich sense of the stochastic term.

We introduce the canonical sample space of Wiener precesses $\Omega: = C_0( {\mathbb{R}})$, the set of continuous functions on ${\mathbb{R}}$ with $0$ at $0$ with the compact open topology. ${\mathscr{F}}$ denotes the Borel $\sigma$-algebra of $\Omega$ and $P$ is the Wiener measure on $(\Omega, {\mathscr{F}})$. The Brownian motion $W(t,\omega)$ is identified as $\omega(t)$, i.e., $W(t,\omega) = \omega(t)$. Furthermore, there is a Wiener shift $\{\vartheta\}_{t\in {\mathbb{R}}}$ on $\Omega$ defined by $\vartheta_t\omega(\cdot) = \omega(t+\cdot)-\omega(t)$ for every $t\in {\mathbb{R}}$ and $\omega\in\Omega$. Then $\vartheta: {\mathbb{R}}\times\Omega\mapsto\Omega$ is $( {\mathcal{B}}( {\mathbb{R}})\times {\mathscr{F}}, {\mathscr{F}})$-measurable with $\vartheta_tP = P$ for all $t\in {\mathbb{R}}$.

Let the mapping $\mathcal{G}_\delta: \Omega\mapsto {\mathbb{R}}$ be a random variable such that

By the Wiener shift $\{\vartheta_t\}_{t\in {\mathbb{R}}}$ on $\Omega$, we get

By the stationary increment property of $\omega(t)$, we have $\mathcal{G}_\delta(\vartheta_t\omega)\thicksim N(0,\frac{1}{\delta})$ for every $\delta\neq0$ and $t\in {\mathbb{R}}$, and moreover it is easy to check that

and $\mathcal{G}_\delta(\vartheta_{t+r}\omega)-\mathcal{G}_\delta(\vartheta_{t}\omega)\thicksim N(0,\frac{2}{\delta})$ for $\delta<r$. Then the process $\mathcal{G}_\delta(\vartheta_t\omega)$ also possesses stationary increment. Furthermore, $\mathcal{G}_\delta(\vartheta_t\omega)$ may be regarded as an approximation of the white noise in the sense that for every $T>0$,

see [21].

Put

In this paper, we study the following point-wise deterministic quasi-linear parabolic equations driven by the process $W_\delta$:

Note that Eq.(5) is a random non-autonomous differential equation. Its solutions admit a non-autonomous random dynamical system and therefore one can study its path-wise dynamical properties such as random attractor and its regular properties.

On the other hand, in terms of the convergence property (4), we will show that the limit of solutions of the deterministic differential equations (5) is a solution of stochastic differential equation (1), which is equivalent to the following Itô stochastic differential equations:

see Theorem 4.9. The upper semi-continuity of their random attractors in $L^2( {\mathbb{R}}^N)$ is also proved in Theorem 4.11. We here remark that the random elements (noise) in the sense of Stratonovich or Itô may propose very different stabilization and destabilization effects on the real models; see [6,5,23].

Our third work in this paper is to establish the Wong-Zakai approximation in the Banach space $L^q(q\geq p>2)$, where $q$ is the growth exponent of the nonlinearity. To this end, some further compactness in $L^q$ is needed. This is achieve by a truncation approach, by which we prove that the solution vanishes in $L^q(q\geq p>2)$ on a domain on which the solution of Eq.(5) diverges to positive and negative infinite. Then by the theorem in [38], we obtain that the random attractor of Eq.(5) converges to that of Eq.(1) in $L^q( {\mathbb{R}}^N)(q\geq p>2)$ in the sense of upper semi-continuity; see Theorem 5.4.

Finally, with some small additional assumptions on the coefficients, by using an induction technique we obtain the solution is bounded in $L^l( {\mathbb{R}}^N)$ for arbitrary $l>q$. By Sobolev interpolation, we establish some compactness of random attractors in $L^l( {\mathbb{R}}^N)$. Following this, we obtain the Wong-Zakai approximations in this higher-regular space $L^l( {\mathbb{R}}^N)$ for arbitrary $l>q$; see Theorem 6.4.

We now recollect some literature on the studying of $p$-Laplacian equation. This equation arises naturally in a boundary value problem of partial differential equations and has been widely used in various fields of science and technology; see [2,7,8] and the references therein. In the absence of noise, that is a deterministic $p$-Laplacian equations, many studies have been done on various aspects of attractors. For the existence of a global attractor in $W^{1,p}\cap L^q$ when the state space is bounded or unbounded, we refer the readers to [11,12,14,32,33]. For the existence and upper semi-continuity of global attractors of $p(x)$-Laplacian systems is studied in [25]. Recent years, the random dynamics of stochastic $p$-Laplacian equation with additive or multiplicative noise, has been intensively investigated in several literature. The existence and upper semi-continuity of random attractors in $L^2( {\mathbb{R}}^N)$ were obtained by Krause et al. [16,17] by using the the well-established theory of pullback random attractors in Wang [27]. Then the regularity and upper semi-continuity were extensively studied in [18,19,20,34]. By estimating the difference of solutions, Zhao [35,37] studied the regularity dynamics in higher regular spaces, where the $L^\delta( {\mathbb{R}}^N)(\forall \delta>2)$-attracting of the random attractor was proved for arbitrary space dimension $N\geq1$.

The approximation of stochastic equations by path-wise deterministic equations was initiated by Wong and Zakai [30,31]. So far, there has been a rich literature about the Wong-Zakai approximations, and we only mention some recent work related to our topic. By means of the Wong-Zakai approximations, Brzeźniak et al.[3] and Manna et al.[22] proved the existence and uniqueness of solution of stochastic Landau-Lifshitz-Gilbert equations with different energy. Lv and Wang et al.[21,29] and Shen et al.[24] studied the approximations of random attractors and invariant manifolds for stochastic partial differential equations. More recently, Sun et al.[26] studied the upper semi-continuity of attractors for the Wong-Zakai approximation of the fractional stochastic reaction-diffusion equation driven by a white noise in $L^2(\mathbb{R}^N)$. Jiang et al.[13] studied the smooth Wong-Zakai approximations given by a stochastic process via Wiener shift and mollifier of Brownian motions. However, there are few papers to attack the Wong-Zakai approximations in higher regular spaces, except that the recent progress in Zhao and Zhang [40,39]. Especially, in [39] we require that the nonlinearity exerted on the equations is monotonic.

This paper is organized as follows. In the next section, we introduce some notions on the random dynamical systems. In section 3, we give the conditions for the coefficients of the stochastic $p$-Laplacian equation, and the related properties of the stationary process $\mathcal{G}_\delta(\vartheta_t\omega)$. Section 4 is devoted to establish the existence of tempered random attractor, the convergence of solutions and the corresponding Wong-Zakai approximation result in $L^2( {\mathbb{R}}^N)$, which will be used later to prove the upper semi-continuity of attractors in $L^q(\mathbb{R}^N)$. In section 5, we obtain the uniform compactness by a truncation approach, where the high-order Wong-Zakai approximations of the stochastic $p$-Laplacian equation driven by multiplicative white noise in $L^q(\mathbb{R}^N),\forall q\geq p>2$ is established. In the final section, we obtain the Wong-Zakai approximation results in $L^l(\mathbb{R}^N)$ for arbitrary $l>q$.

2.   Preliminaries on non-autonomous random attractors

We present in this section some basic notions about (non-autonomous) random attractor $\mathcal{A}(t,\omega)$ [15,28], which is a generalization of the (autonomous) random attractor $\mathcal{A}(\omega)$ for random dynamical system; see [10,9]. For a comprehensive knowledge of random dynamical systems, the reader may refer to Arnold [1]. Let $(H,\|\cdot\|_H)$ be a separable Banach space with $\sigma$-algebra $\mathfrak{B}(X)$ and $\vartheta = (\Omega, {\mathscr{F}},P,\{\vartheta_{t}\}_{t\in\mathbb{R}})$ be a metric dynamical system(MDS); see [1]. Denote by ${\mathbb{R}}^+ = \{s\geq 0: s\in {\mathbb{R}}\}$.

Definition 2.1. A mapping $\varphi:\mathbb{R}^{+}\times\mathbb{R}\times\Omega\times X\mapsto X$ is called a random cocycle on $X$ over an MDS $\vartheta$ if for all $\tau\in\mathbb{R}$, $\omega\in\Omega$ and $t,s\in\mathbb{R}^+$,

(ⅰ) $\varphi(\cdot,\tau,\cdot,\cdot):\mathbb{R}^{+}\times\Omega\times X\mapsto X$ is $(\mathfrak{B}(\mathbb{R}^+)\times {\mathscr{F}}\times\mathfrak{B}(X),\mathfrak{B}(X))$-measurable;

(ⅱ) $\varphi(0,\tau,\omega,\cdot)$ is the identity on $X$;

(ⅲ) $\varphi(t+s,\tau,\omega,\cdot) = \varphi(t,\tau+s,\vartheta_s\omega,\varphi(s,\tau,\omega,\cdot))$.

A random cocycle $\varphi$ is said to be continuous in $X$ if for each $t\in\mathbb{R}^+$, $\tau\in\mathbb{R}$ and $\omega\in\Omega$, the mapping $\varphi(t,\tau,\omega,\cdot):X\mapsto X$ is continuous.

Definition 2.2. A family of sets $K = \{K(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}$ is called a random set in $X$ with respect to ${\mathscr{F}}$ if the mapping $\omega\in\Omega\mapsto \text{dist}_X(\{x\},K(\tau,\omega))$ is (${\mathscr{F}},\mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in X$ and $\tau\in \mathbb{R}$, where $\text{dist}_X$ is the Hausdorff semi-metric in $2^X$, i.e., for two nonempty subsets $A,B\in 2^X$, $\text{dist}_X(A,B) = \sup_{a\in A}\inf_{b\in B}\|a-b\|_X.$

Throughout this paper, we denote by ${\mathfrak{D}}$ a collection of some families of non-empty subsets of $X$: ${\mathfrak{D}} = \{D(\tau,\omega)\subset X: D(\tau,\omega)\neq\emptyset,\tau\in {\mathbb{R}},\omega\in\Omega\}.$ The elements $D_1$ and $D_2$ of ${\mathfrak{D}}$ is said to be equal if $D_1(\tau,\omega) = D_2(\tau,\omega)$ for all $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$. ${\mathfrak{D}}$ is said to be inclusion closed if for $D = \{D(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}\in {\mathfrak{D}}$ and $D_0(\tau,\omega)\subset D(\tau,\omega)$ for every $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$, then $D_0 = \{D_0(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}\in {\mathfrak{D}}$ (see [28]).

Definition 2.3. Let ${\mathfrak{D}}$ be a collection of some families of non-empty subsets of $X$. A family of sets $K = \{K(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}$ is called a $\mathfrak{D}$-pullback absorbing set for $\varphi$ in $X$ if $K\in {\mathfrak{D}}$ and for every $\tau\in\mathbb{R},\omega\in\Omega$ and $D = \{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}\in {\mathfrak{D}}$, there exists a time $T = T(\tau,\omega,D)>0$ such that for all $t\geq T$,

Furthermore, if $K$ is a random set then $K$ is called a $\mathfrak{D}$-pullback random absorbing set for $\varphi$.

Definition 2.4. A family of sets $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}$ is called a $\mathfrak{D}$-pullback random attractor for $\varphi$ in $X$ if $\mathcal{A}\in\mathfrak{D}$ is random set and for every $\tau\in\mathbb{R}$ and $\omega\in\mathfrak{D}$, the following three conditions hold:

(ⅰ) $\mathcal{A}(\tau,\omega)$ is compact in $X$;

(ⅱ) $\mathcal{A}(\tau,\omega)$ is invariant, that is, $\varphi(t,\tau,\omega,\mathcal{A}(\tau,\omega)) = \mathcal{A}(\tau+t,\vartheta_t\omega)$, for arbitrary $t\geq0$;

(ⅲ) $\mathcal{A}(\tau,\omega)$ is attracting in $X$, that is, for every element $D\in\mathfrak{D}$,

Definition 2.5. Let ${\mathfrak{D}}$ be a collection of some families of non-empty subsets of $X$. A random cocycle $\varphi$ is said to be $\mathfrak{D}$-pullback asymptotically compact in $X$ if for all $\tau\in\mathbb{R},\omega\in\Omega$ and $\forall t_n\rightarrow +\infty, x_n \in D(\tau-t_n,\vartheta_{-t_n}\omega)$ with $D\in {\mathfrak{D}}$, the sequence

Theorem 2.6 ([28]). Let ${\mathfrak{D}}$ be a collection of some families of non-empty subsets of $X$ and $\varphi$ be a continuous random cocycle on $X$ over an MDS $\vartheta$. Then $\varphi$ has a unique $\mathfrak{D}$-pullback random attractor $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ in $X$ if $\varphi$ has a closed $\mathfrak{D}$-pullback random absorbing set $K = \{K(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ in $X$ and $\varphi$ is $\mathfrak{D}$-pullback asymptotically compact in $X$. Furthermore, for all $\tau\in\mathbb{R},\omega\in\Omega$,

which is the omega-limit set of $K(\tau,\omega)$.

Theorem 2.7 ([19,38]). Let $I$ be a metric space, $\delta,\delta_0\in I$, and $\varphi_\delta$ be a continuous cocycle on $X$. Suppose that

(i) For every $t\in {\mathbb{R}}^+$, $\tau\in {\mathbb{R}}$, $\omega\in\Omega$, $\delta_n\rightarrow \delta_0$ with $\delta_n\in I$ and $x_n,x\in X$ with $x_n\rightarrow x$, there holds

(ii) Let $\mathfrak{D}$ be a collection of some families of nonempty subsets of $X$, for every $\tau\in {\mathbb{R}}, \omega\in\Omega$ there exist $\varrho_{\delta_0}(\tau, \omega)>0$ such that

Let $\mathcal{A}_\delta$ be a $\mathfrak{D}$-pullback attractor and $K_\delta$ a $\mathfrak{D}$-pullback absorbing set of $\varphi_\delta$ in $X$, such that for all $\tau\in {\mathbb{R}}, \omega\in\Omega$,

(iii) For all $\tau\in {\mathbb{R}}, \omega\in\Omega$, if $\delta_n\rightarrow \delta_0$ and $x_n\in \mathcal{A}_{\delta_n}(\tau,\omega)$, then $x_n$ is pre-compact in $X$.

Then for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$, we have

In addition, for all $\tau\in {\mathbb{R}}, \omega\in\Omega$, if $\delta_n\rightarrow \delta_0$ and $x_n\in \mathcal{A}_{\delta_n}(\tau,\omega)$, then $x_n$ is pre-compact in $Y$. Then for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$, we have

where we assume that $\varphi(t,\tau,\omega,\cdot): X\mapsto Y$ for every $t>0$, $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$, and both $Y$ and $X\cap Y$ are Banach spaces.

3.   Mathematical settings

In this article, we denote by $c,c_i>0(i = 1,2,...)$ the generic constants which may depend on the $\tau,\omega$ and $T$, and may change their values from to line to line. Denote by $\|\cdot\|_p (p>0)$ the norm in $L^p(\mathbb{R}^N)$ and in particular $\|\cdot\| = \|\cdot\|_2$ for $p = 2$, and $\|\cdot\|_{W^{1,p}}$ the norm in $W^{1,p}( {\mathbb{R}}^N)$.

In this section, we give some mathematical settings of Eq.(1), including the conditions on the nonlinearity $f$ and some known results in [21,29]. We assume that the nonlinear function $f$ is continuous on $\mathbb{R}\times\mathbb{R}^N\times\mathbb{R}$ and satisfies the following conditions: for all $t,s\in\mathbb{R}$ and $x\in\mathbb{R}^N$,

where $2<p\leq q, \alpha_{1}>0, \alpha_{2}>0$, $\psi_{1} \in L_{l o c}^{1}(\mathbb{R}; L^{1}(\mathbb{R}^{N}))$, $\psi_{2} \in L_{l o c}^{q_{1}}(\mathbb{R}; L^{q_{1}}(\mathbb{R}^{N}))$ with $\frac{1}{q_1}+\frac{1}{q} = 1$, and $\psi_{3} \in L_{l o c}^{\infty}(\mathbb{R}; L^{\infty}(\mathbb{R}^{N}))$.

In addition, the following condition will be needed for the non-autonomous terms $g$ and $\psi_1$ when deriving uniform estimates of solutions:

where $\lambda$ is as in Eq.(1), and especially in order to derive the tempered property of attractor we further assume that for arbitrary $c>0$,

Let $D = \{D(\tau,\omega)\subset L^2(\mathbb{R}^N):\tau\in\mathbb{R},\omega\in\Omega\}$ be a family of bounded nonempty subsets of $L^2(\mathbb{R}^N)$. Such a family $D$ is called tempered if for every $c>0,\tau\in\mathbb{R}$ and $\omega\in\Omega$,

where $\|D\| = \sup_{u\in D}\|u\|$. In what follows, we always suppose that $\mathfrak{D}$ is a collection of all families of tempered subsets of $L^2(\mathbb{R}^N)$, namely,

Then it is obvious that $\mathfrak{D}$ is inclusion closed.

For convenience, let us present the uniform convergence of the integral of stationary process $\mathcal{G}_\delta(\vartheta_t\omega)$ on any finite interval, which is stated in [21].

Lemma 3.1. Given $\tau\in\mathbb{R},\omega\in\Omega$ and $T>0$, then for every $0<\varepsilon<1$, there exists $\delta_0 = \delta_0(\varepsilon,\tau,\omega,T)>0$ such that for all $0<|\delta|\leq \delta_0$,

By the continuity of $\omega(t)$ on $[\tau,\tau+T]$ for any $\tau\in\mathbb{R}$ and $T>0$, there exists $c = c(\tau,\omega,T)$ such that

By (13)-(14) there are positive constants $\delta_1 = \delta_1(\tau,\omega,T)$ and $c = c(\tau,\omega,T)$ such that for all $0<|\delta|\leq\delta_1$,

We give a convenient lemma about the property of the stationary process $\mathcal{G}_\delta(\vartheta_t\omega)$, which will be used repeatedly in the subsequential arguments.

Lemma 3.2. Let $\tau\in {\mathbb{R}}$, $\omega\in\Omega, \delta\neq0$ and $\mathcal{G}_\delta(\vartheta_t\omega)$ be defined by (3). Then for any $\gamma>0$, there exist positive constants $\widetilde{c} = \widetilde{c}(\tau,\omega,\lambda)$ and $\delta_0 = \delta_0(\tau,\omega)<1$ such that for any $s\leq 0$ and $0<|\delta|\leq\delta_0$,

Proof. By the definition (2) of $\mathcal{G}_{\delta}$, we have

Then according to the integration mean theorem, there exists $r_0$ between $s$ and $s+\delta$ such that

Since the Wiener process $\omega(t)$ satisfies $\lim\limits_{|t|\rightarrow \infty}\frac{\omega(t)}{t} = 0$, there exists $T_1 = T_1(\omega,\lambda,\gamma)\leq0$ such that for all $r_0\leq T_1$ we have $|\omega(r_0)|\leq -\min{\{\frac{\lambda}{6}, \frac{\gamma}{8}}\}r_0$ for any $\gamma>0$. Note that $r_0-s\leq |\delta|$. Then if $s\leq T_1-|\delta|$ then $r_0\leq T_1$, whence we know that for all $s\leq T_1-|\delta|$,

Consider that $s-r_0\leq |\delta|$ also holds true. Then by (18), for all $0<|\delta|\leq1$ and $s\leq T_1-1$,

On the other hand, we have $T_1+\delta-1\leq s+\delta\leq \delta$ for $T_1-1\leq s\leq 0$. Then the number $r_0$ in (17) may belong to a finite closed interval $[-|T_1|-|\delta|-1,|T_1|+|\delta|+1]$ for $T_1-1\leq s\leq 0$, and therefore $r_0\in[-|T_1|-2,|T_1|+2]$ for $0<|\delta|\leq1$ and $T_1-1\leq s\leq 0$. Then by the continuity of $\omega$ again, there exists $c_2(\omega)$ such that for all $T_1-1\leq s\leq 0$ and $0<|\delta|\leq1$,

By (19) and (20) we get for all $0<|\delta|\leq1$ and $s\leq0$,

According to the continuity of $\omega(t)$ in $t$, we have $\lim\limits_{\delta\rightarrow0}\frac{1}{\delta}\int_{-\tau+\delta}^{-\tau}\omega(r)dr = \omega(-\tau)$, and thereby there exists $\delta_1 = \delta_1(\tau,\omega)$ such that for all $0<|\delta|\leq\delta_1$,

Let $\delta_0 = \min\{\delta_1,1\}$. Then from (16), (21) and (22) we obtain that, for all $0<|\delta|\leq\delta_0$ and $s\leq0$,

which completes the proof.

To discuss the random attractors and the Wong-Zakai approximation, we need to transform the stochastic $p$-Laplacian equation (1) into a path-wise deterministic equation. We first make a transformation for Eq.(1) by means of the geometric Brownian motion $e^{\omega(t)}$. By Itô' formula (see e.g.[4,Theorem 6.2.1]), we have

Let $u(t,\tau,\omega,u_\tau) = e^{\omega(t)}v(t,\tau,\omega,v_\tau)$, where $u$ is the solution of Eq.(6). Then we have

in the Itô sense. Therefore $v$ solves the following equation: for any $\tau\in\mathbb{R}$, $\omega\in\Omega$, $t>\tau$ and $x\in {\mathbb{R}}^N$,

with the initial condition $v(\tau,x) = v_\tau$.

We also introduce the transformation:

with $v_{\delta,\tau} = e^{-\int_{0}^{\tau}\mathcal{G}_\delta(\vartheta_s\omega)ds}u_{\delta,\tau}$. Then we get from Eq.(5) that for any $t>\tau$ and $x\in {\mathbb{R}}^N$,

with the initial condition $v_\delta(\tau,x) = v_{\delta,\tau}$.

Give $\omega \in \Omega, \tau \in \mathbb{R}$ and $v_{\tau} \in L^{2}(\mathbb{R}^{N})$, if $f$ satisfies (7)-(9) then Eq.(24) has a unique solution $v(\cdot, \tau, \omega, v_{\tau}) \in C([\tau, \infty); L^{2}(\mathbb{R}^{N}))\cap L_{loc}^q((\tau,\infty); L^q( {\mathbb{R}}^N))$. In addition $v(\cdot, \tau, \omega, v_{\tau})$ is continuous in the initial $v_{\tau}$ in $L^{2}(\mathbb{R}^{N})$ and is $( {\mathscr{F}},\mathcal{B}(L^{2}(\mathbb{R}^{N})))$-measurable in $\omega \in \Omega$, and thus we can define a continuous cocycle $\varphi :\mathbb{R}^{+} \times \mathbb{R}\times \Omega \times L^{2}(\mathbb{R}^{N}) \rightarrow L^{2}(\mathbb{R}^{N})$ for Eq.(1) by

for every $t\in {\mathbb{R}}^+, \tau \in \mathbb{R}$ and $\omega \in \Omega$. By a similar method, we define the random cocycle for Eq.(5) by

4.   Random attractors and $L^2$-Wong-Zakai approximations

In this section, we consider the existence of random attractors and the Wong-Zakai approximations of $p$-Laplacian equation in $L^2( {\mathbb{R}}^N)$.

4.1.   Existence of random attractors

In this subsection, we present the existence of random attractors for random cocycles defined by (27) and (28) without detailed proof.

Lemma 4.1. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Then the continuous cocycle $\varphi$ associated with Eq.(1) admits a closed $\mathfrak{D}$-pullback random absorbing set $K = \{K(\tau, \omega): \tau \in \mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$ in $L^2( {\mathbb{R}}^N)$, which is depicted by, for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $\varrho(\tau,\omega)$ is given by

Proof. Taking the inner product of (24) with $v$ in $L^{2}(\mathbb{R}^{N})$, we have

Using the assumptions (7), we get

For the $p$-Laplacian part in (31), we have

The second term on the right hand side of (31) is bounded by

Combine (31)-(34) to find

Multiplying (35) by $e^{\frac{4\lambda}{3}t}$ and integrating from $\tau-t$ to $\xi$, along with replacing $\omega$ by $\vartheta_{-\tau}\omega$, and then using the formula $\vartheta_{-\tau}\omega(s) = \omega(s-\tau)-\omega(-\tau)$, we find that

By the formula $u(\xi,\tau-t,\vartheta_{-\tau}\omega,u_{\tau-t}) = e^{\omega(\xi-\tau)-\omega(-\tau)}v(\xi,\tau-t,\vartheta_{-\tau}\omega,v_{\tau-t})$, we have

Let $\xi = \tau$. Then we get

Since $u_{\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$, then there exists a $T = T(\tau,\omega,D)$ such that for all $t\geq T$, $e^{-\frac{4\lambda}{3} t-2\omega(-t)}\|u_{\tau-t}\|^{2}\leq 2\int_{-\infty}^{0}e^{\frac{4\lambda}{3} s-2\omega (s)}(\frac{3}{2\lambda}\|g(s+\tau)\|^{2}+\|\psi_{1}(s+\tau)\|_{1})ds$. From (10) we can deduce that $\int_{-\infty}^{0}e^{\frac{4\lambda}{3} s-2\omega (s)}(\frac{3}{2\lambda}\|g(s+\tau)\|^{2}+\|\psi_{1}(s+\tau)\|_{1})ds<+\infty$. Denote by

Then by (11) it is easy to check that $K(\tau,\omega) = \{u\in L^2( {\mathbb{R}}^N): \|u\|^2\leq \varrho(\tau,\omega)\}$ is tempered; that is, $\lim\limits_{t\rightarrow -\infty} e^{\gamma t}\|K(\tau+t,\vartheta_t\omega)\|^2\leq \lim\limits_{t\rightarrow -\infty} e^{\gamma t}\varrho(\tau+t,\vartheta_t\omega) = 0$ for any $\gamma>0$. Furthermore, since for fixed $\tau\in {\mathbb{R}}$ and for each $\omega\in\Omega$ and $x\in L^2( {\mathbb{R}}^N)$,

then the mapping $\omega\mapsto dist(x, K(\tau,\omega))$ is (${\mathscr{F}},\mathcal{B}(\mathbb{R}))$-measurable for every fixed $x\in X$, and therefore $K(\tau,\omega) = \{u\in L^2( {\mathbb{R}}^N): \|u\|^2\leq \varrho(\tau,\omega)\}$ is a closed random set, which completes the proof.

To prove the $\mathfrak{D}$-pullback asymptotical compactness in $L^2( {\mathbb{R}}^N)$ for Eq.(1), we need to prove that the tail of solution is small enough on larger and larger space domains, which is given as below.

Lemma 4.2. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Given $\tau \in \mathbb{R}, \omega \in \Omega$, $D = \{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$, then for every $u_{\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ and any $\varepsilon>0,$ there exist $T = T(\tau, \omega, D, \varepsilon) \geq 1$ and $R = R(\tau,\omega,\varepsilon)>0$ such that for all $\xi\in[\tau-1,\tau]$, the solution $u$ of Eq.(1) satisfies

Proof. Let $\rho$ be a smooth function defined on $\mathbb{R}^+$ such that $0\leq\rho(s)\leq1$ for all $s\in\mathbb{R}^+$,

Then there exists a positive constant $c$ such that $|\rho^\prime(s)|\leq c_0$ for all $s\in\mathbb{R}^+$. For convenience, we write $\rho(\frac{|x|^2}{k^2}) = \rho_k$. Taking the inner product of Eq.(24) with $\rho_kv$ in $L^2(\mathbb{R}^N)$, we obtain that

By (7), the first term on the right-hand side of (40) satisfies

For the forcing term we have

For the $p$-Lapacian part, by using Young's inequality and Sobolev interpolation inequality $\|u\|^p_p\leq \|u\|^q_q+\|u\|^2 (q\geq p>2)$, we have

where $c_1 = c_1(p,q,c_0)$. It follows from (40)-(43) that

Applying Gronwall's lemma and replacing $\omega$ by $\vartheta_{-\tau}\omega$, we have for every $\xi\in[\tau-1,\tau]$,

Since $u_{\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ and $D$ is tempered, we find that for every $\xi\in[\tau-1,\tau]$,

As a consequence, there exists a $T_1 = T_1(\tau,\omega,D)$ such that for all $t\geq T_1$,

Using (10), there exists a radius $R_1 = R_1(\tau,\omega,\varepsilon)$ such that for every $\xi\in[\tau-1,\tau]$ and all $k\geq R_1$,

From (37) it follows that there exists a $T_2 = T_2(\tau,\omega,D)\geq T_1$ such that for all $t\geq T_2$,

Then there is a radius $R_2 = R_2(\tau,\omega,D,\varepsilon)\geq R_1$ such that for every $\xi\in[\tau-1,\tau]$ and all $t\geq T_2$ and $k\geq R_2$,

Therefore it follows from (45)-(48) that for every $\xi\in[\tau-1,\tau]$ and all $t\geq T_2(\tau,\omega,D)$ and $k\geq R_2$,

which along with the formula

implies the desired result.

We now prove the $\mathfrak{D}$-pullback asymptotical compactness of solutions of Eq.(1) in $L^2( {\mathbb{R}}^N)$.

Lemma 4.3. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Then the continuous cocycle $\varphi$ defined by (27) is $\mathfrak{D}$-pullback asymptotically compact in $L^{2}( {\mathbb{R}}^N)$.

Proof. This is followed by a same procedure as in [39,Lemm 3.6].

By Lemmas 4.1, 4.3 along with Theorem 2.6 we immediately get the following result.

Theorem 4.4. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Then the continuous cocycle $\varphi$ generated by the solution of Eq.(1) admits a unique tempered $\mathfrak{D}$-pullback random attractor $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}\in\mathfrak{D}$ in $L^2( {\mathbb{R}}^N)$ which is structured by, for each $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $K = \{K(\tau,\omega): \tau\in R,\omega\in\Omega\}$ is given in Lemma 4.1.

Proof. It follows from Lemma 4.1 that there exists $T = T(\tau,\omega,D)>0$, such that for all $t\geq T$,

therefore, $K\in\mathfrak{D}$ is a closed $\mathfrak{D}$-pullback random absorbing set of $\varphi$, and by Lemma 4.3 $\varphi$ is $\mathfrak{D}$-pullback asymptotically compact in $L^2( {\mathbb{R}}^N)$. Then the desired result follows from Theorem 2.6 immediately.

In what follows, we will prove the existence of ${\mathfrak{D}}$-pullback random absorbing set $K_\delta$ for the cocycle $\varphi_\delta$ in $L^2( {\mathbb{R}}^N)$.

Lemma 4.5. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Then the continuous cocycle $\varphi_{\delta}$ associated with Eq.(5) has a closed $\mathfrak{D}$-pullback random absorbing set $K_{\delta} = \{K_{\delta}(\tau, \omega):\tau\in\mathbb{R},\omega\in\Omega\}\in\mathfrak{D}$, which is structured by, for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $\varrho_{\delta}(\tau, \omega)$ is given by

In addition, for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$, $\lim_{\delta \rightarrow 0}\varrho_{\delta}(\tau, \omega) = \varrho(\tau, \omega)$, where $\varrho(\tau,\omega)$ is defined by (30).

Proof. By a similar technique as in the proof of Lemma 4.1 we have

In (52), utilizing Gronwall's lemma over the interval $[\tau-t,\xi]$ with $\xi\in[\tau-1,\tau],t\geq 1$, we get

It follows from (53) and the formula

that

Let $\xi = \tau$ in (54). Then for every $u_{\delta,\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ there exists $T_1 = T_1(\tau,\omega,\delta,D)>2$ such that for all $t\geq T_1$,

That is, for all $t\geq T_1$,

By (55), we can verify that $K_{\delta}(\tau, \omega)$ is tempered. Indeed, for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$, it follows from (51) that for any $\gamma>0$,

Note that by Lemma 3.2, there exist positive constants $\tilde{c} = \tilde{c}(\omega)$ and $\delta_0 = \delta_0(\omega)$ such that for every $s\leq0$ and $0<|\delta|\leq\delta_0$,

Then by (58) for every $s\leq0,t\leq0$ and $0<|\delta|\leq\delta_0$,

From (57) and (59) it gives that

Consequently, by (11) and (60) we have for any $\gamma>0$ and $0<|\delta|\leq\delta_0$,

On the other hand, by Lemma 3.1 and Lebesgue'dominated convergence theorem, we have

for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$.

Furthermore, since by (58) we get $e^{\frac{4\lambda}{3}(s-\tau)-2\int_{0}^{s-\tau}\mathcal{G}_\delta(\vartheta_{r}\omega)dr}\geq e^{-\widetilde{c}+\frac{5}{3}\lambda(s-\tau)}$ for every $s\leq\tau$ and $0<|\delta|\leq\delta_0$, then it follows that for every $0<|\delta|\leq\delta_0$, the second term on the left hand side of (55) is bounded by

which concludes the proof.

Lemma 4.6. Suppose (7)-(11) hold. Given $\tau\in\mathbb{R}, \omega\in\Omega$, then for every $\varepsilon>0$, there exist $\delta_{0} = \delta_{0}(\tau,\omega,\varepsilon)>0, T = T(\tau, \omega, \varepsilon)>0$ and $R = R(\tau, \omega, \varepsilon)>0$ such that for all $0<|\delta|\leq\delta_{0}$, the solution of Eq.(5) satisfies

for all $u_{\delta, \tau-t} \in K_{\delta}\left(\tau-t, \vartheta_{-t} \omega\right)$, where $K_{\delta}$ is given by (50).

Proof. By some similar calculations as in (44), from (26) we get

Multiply (63) by $e^{\frac{4\lambda}{3}t}$ and then integrate over the interval $[\tau-t,\tau]$, along with $\omega$ replaced by $\vartheta_{-\tau}\omega$, to find that

from which it follows that

Since $u_{\delta,\tau-t}\in K_\delta(\tau-t,\vartheta_{-t}\omega)$, then by (58) we have for all $0<|\delta|\leq\delta_0$ and $t\geq0$,

from which and (61) that there exists $T_1 = T_1(\tau,\omega,\varepsilon)>0$ such that for all $t\geq T_1$ and $0<|\delta|\leq\delta_0$,

By (54), (58) and (10), there exists $T_2 = T_2(\tau,\omega,\varepsilon)\geq T_1$ such that for all $t\geq T_2$ and $0<|\delta|\leq\delta_0$,

by which it gives that there exists a constant $R_1 = R_1(\tau,\omega,\varepsilon)>0$ such that for all $t\geq T_2, k\geq R_1$ and $0<|\delta|\leq\delta_0$,

By (58) and (10), there exists $R_2 = R_2(\tau,\omega,\varepsilon)\geq R_1$ such that for all $k\geq R_2$ and $0<|\delta|\leq\delta_0$,

Then combine (64)-(67) to get that for all $k\geq R_2$ and $0<|\delta|\leq\delta_0$,

which completes the proof.

Consider that if we let $\sigma(t,x,u) = u$ then $f$ and $\sigma(t,x,u)$ satisfy the total assumptions (8)-(12) in the reference [39]. Thus by Theorem 3.7 in that paper the random cocycle $\varphi_\delta(\delta\neq0)$ defined by (28) possesses a unique random attractor in $L^2( {\mathbb{R}}^N)$, which reads as follows.

Theorem 4.7. Suppose (7)-(11) hold and $\mathfrak{D}$ is defined by (12). Then the continuous cocycle $\varphi_\delta$ defined by (28) admits a unique $\mathfrak{D}$-pullback random attractor in $L^2( {\mathbb{R}}^N)$, which is pictured by, for each $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $K_\delta = \{K_\delta(\tau,\omega): \tau\in {\mathbb{R}},\omega\in\Omega\}$ is defined in Lemma 4.5.

4.2.   Wong-Zakai approximation in $L^2$

In this subsection, we study the Wong-Zakai approximation of solutions in $L^2( {\mathbb{R}}^N)$. We first prove the convergence of solutions to Eq.(5).

Lemma 4.8. Suppose (7)-(11) hold. Given $\tau\in\mathbb{R},\omega\in\Omega, T>0$ and $\varepsilon\in(0,1)$, let $u$ and $u_{\delta}$ be the solutions of Eq.(1) and (5) with initial data $u_{\delta, \tau}$ and $u_{\tau}$, respectively. Then there exist $\delta_{0} = \delta_{0}(\tau, \omega, T, \varepsilon)>0$ and $c = c(\tau, \omega, T)>0$ such that

Proof. Let $V_{\delta}(t,\tau,\omega,V_{\delta,\tau}) = v_{\delta}(t)-v(t) = v_{\delta}(t,\tau,\omega,v_{\delta,\tau})-v(t,\tau,\omega,v_{\tau})$, where $v(t)$ and $v_\delta(t)$ are the solutions of (24) and (26), respectively. Then $V_{\delta}$ satisfies that for all $t\geq\tau$,

with the initial $V_{\delta}(\tau,x) = V_{\delta,\tau} = v_{\delta,\tau}-v_{\tau}$. From (70), we have

Put

Then we have

By Lemma 3.1 and (14)-(15), we deduce that for arbitrary $0<\varepsilon<1$, there exists $c_0 = c_0(\tau,\omega,T)$ and $\delta_0 = \delta_0(\tau,\omega,T,\varepsilon)$ such that for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$,

and

We are now ready to estimate the terms in (71). We rewrite the nonlinearity as

By (72), (8) and (9), the first term on the right hand side of (77) is estimated as

where $\frac{1}{q_1}+\frac{1}{q} = 1$, for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$, and the second term on the right hand side of (77), using (8) and (74), we get

for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$. By a combination of (77)-(79) we find that for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$,

where $c = c(\tau,\omega,T)$. For the $p$-Laplacian part we have

where the first term on the right hand side of (81) is estimated as

since the mapping $s\mapsto |s|^{p-2}s$ for $p\geq2$ is increasing on ${\mathbb{R}}$. The second term on the right hand side of (81), by (75) we have

By a combination of (81)-(83) we get that for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$,

By (74) and Young's inequality, we have for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$,

Therefore, plugging (80) and (84)-(85) into (71), it follows that for all $0<|\delta|\leq\delta_0$ and $t\in[\tau, \tau+T]$,

where we have used

Apply Gronwall's lemma in (86) over the interval $[\tau,\tau+T]$ to show that for all $0<|\delta|\leq\delta_{0}$ and $t \in[\tau, \tau+T]$,

Integrate (35) from $\tau$ to $t$ to yield

Similarly, by (52) we have

Then (88)-(90) together imply that

from which and (87) we get for all $0<|\delta|\leq\delta_{0}$ and $t \in[\tau, \tau+T]$,

This concludes the proof.

From Lemma 4.8, we immediately get the convergence of solutions in $L^2( {\mathbb{R}}^N)$ whenever $\delta_n\rightarrow0$ and $\|u_{\delta_n,\tau}-u_\tau\|\rightarrow0$ as $n\rightarrow \infty$.

Theorem 4.9. Given $\tau\in {\mathbb{R}}, \omega\in\Omega$ and $T>0$, if $\delta_n\rightarrow0$ and $\|u_{\delta_n,\tau}-u_\tau\|\rightarrow0$ as $n\rightarrow \infty$, then for every $t \in[\tau, \tau+T]$,

Next, we derive the compactness result, which is one of the crucial conditions to prove the upper semi-continuity of attractor $\mathcal{A}_{\delta} = \{\mathcal{A}_{\delta}(\tau, \omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$.

Lemma 4.10. Suppose (7)- (11) hold. Then for every $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$, if $\delta_n \rightarrow0$ as $n\rightarrow \infty$ and $u_n\in\mathcal{A}_{\delta_{n}}(\tau,\omega)$, then $\{u_n\}^{\infty}_{n = 1}$ is pre-compact in $L^2(\mathbb{R}^N)$ for $q\geq p>2$.

Proof. By Lemma 4.6 and Theorem 4.9, and a similar procedure to prove [39,Lemma 4.11], we can obtain this result and so the detailed proof is omitted.

We now show that the random attractor of the approximation equation (5) converges to that of the stochastic $p$-Laplacian equation (1) driven by multiplicative white noise in $L^2( {\mathbb{R}}^N)$, in the sense of upper semicontinuity of sets. The main result of this section is given below.

Theorem 4.11. Suppose (7)-(11) hold. Then for every $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$,

where $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ and $\mathcal{A}_\delta = \{\mathcal{A}_\delta(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ are random attractors of Eqs.(1) and (5), respectively.

Proof. Let $\delta_n \rightarrow0$ and $u_{n,\tau}\rightarrow u_{\tau}$ in $L^2(\mathbb{R}^N)$, then by Lemma 4.9 we have that for all $\tau\in {\mathbb{R}}, t\geq 0$ and $\omega\in\Omega$, $\varphi_{\delta_n}(t,\tau,\omega,u_{\delta_n,\tau})\rightarrow \varphi(t,\tau,\omega,u_{\tau})$ in $L^2(\mathbb{R}^N)$; By Lemma 4.5 we have for all $\tau\in {\mathbb{R}}$ and $\omega\in\Omega$, $\limsup_{\delta\rightarrow 0}\|K_{\delta}(\tau, \omega)\|\leq\varrho(\tau, \omega)$. Then in conjunction with Lemma 4.10 we show that all conditions in Theorem 2.7 are fulfilled, and whence the proof is concluded.

5.   $L^q$-Wong-Zakai approximations

In this section, we discuss the convergence of random attractors in $L^q( {\mathbb{R}}^N)$, where $q>2$ is the growth exponent of the nonlinearity. For this purpose, we need to further assume that $\psi_1\in L_{loc}^\frac{q}{2}( {\mathbb{R}}; L^{\frac{q}{2}}( {\mathbb{R}}^N))$ such that

5.1.   Uniform truncation estimates of solutions

We now deal with the uniform $L^q$-estimate of solution of Eq.(26) by a truncation approach. To this end, we need to prove the bound of solutions in $L^q( {\mathbb{R}}^N)$.

Lemma 5.1. Suppose that (7)-(10) and (93) hold and ${\mathfrak{D}}$ is defined by (12). Given $\tau\in\mathbb{R}$ and $\omega\in\Omega$, then for every $u_{\delta,\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ with $D\in {\mathfrak{D}}$, there exist positive constants $\delta_0 = \delta_0(\tau,\omega)$, $\widetilde{c}_i = \widetilde{c}_i(\tau,\omega)(i = 1,2)$ and $T = T(\tau,\omega)\geq2$ such that the solution of Eq.(26) satisfies that

where $v_{\delta,\tau-t} = e^{-\omega(-t)+\omega(-\tau)}u_{\delta,\tau-t}$.

Proof. Taking the inner product of Eq.(26) in $L^2(\mathbb{R}^N)$ with $|v_{\delta}|^{q-2}v_{\delta}$, we have

The first term on the right hand side of (96), we have

By (7), we get

On the other hand, we have

By a combination of (96)-(99), we get

We apply [36,Lemma 6.1] in (100) over the interval $[\tau-2,\xi]$ for $\xi\in[\tau-1,\tau]$ and $p>2$, to find that

We estimate every term on the right hand side of (101). By (62) we deduce that there exist $T = T(\tau,\omega)>0$ and $\delta_1 = \delta_1(\tau,\omega)>0$ such that for all $t\geq T$ and $0<|\delta|\leq\delta_1$,

By Lemma 3.2, there exists $\delta_2 = \delta_2(\tau,\omega)$ such that for all $0<|\delta|\leq\delta_2$, we have $e^{\frac{4\lambda}{3} s-q\int_{-\tau}^{s}\mathcal{G}_\delta(\vartheta_r\omega)dr}\leq e^{\frac{4\lambda}{3} s+qc(\tau,\omega) -\frac{q\lambda}{6}s}\leq e^{qc(\tau,\omega)+\frac{q\lambda}{3}}$ for all $s\in[-2,0]$, and therefore by (10) and (93) it follows that

Let $\delta_0 = \min\{\delta_1,\delta_2\}$. Then (101)-(103) together imply that there exists positive constant $\widetilde{c}_i = \widetilde{c}_i(q,\tau,\omega)(i = 1,2)$ such that

and

which completes the proof.

The following lemma is concerned with the truncation approach, by which we show that the solution of Eq.(5) vanishes in $L^q$ with a changing domain on which its value diverges to infinite.

Lemma 5.2. Suppose that (7)-(10) and (93) hold. Let $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $u_{\tau-t}\in K_\delta(\tau-t,\vartheta_{-t}\omega)$. Then for any $\varepsilon>0$, there exist positive constants $\delta_0 = \delta_0(\tau,\omega,\varepsilon), c = c(\tau,\omega), M_0 = M_0(\tau,\omega,\varepsilon)$ and $T = T(\tau,\omega, \varepsilon)$ such that the solution $u_\delta$ of Eq.(1) satisfies that

where $(|u_\delta(\tau)|\geq M_0) = \{x\in\mathbb{\mathbb{R}^N}: |u_\delta(\tau,\tau-t,\vartheta_{-\tau}\omega,{u}_{\delta,\tau-t})|\geq M_0\}$ and $K_\delta$ is as in (29).

Proof. Given $\tau\in {\mathbb{R}}, \omega\in\Omega$ and $s\in[\tau-1,\tau]$, let $v_\delta(s) = v_\delta(s,\tau-t,\vartheta_{-\tau}\omega,v_{\delta,\tau-t})$ be the solution of Eq.(26) at the sample $\vartheta_{-\tau}\omega$ with the initial value $v_{\delta,\tau-t}$. Put $M = M(\tau,\omega)>0$. Denote by $(v_\delta(s)-M)_+$ the nonnegative part of $v_\delta(s)-M$. Multiplying (26) by $|(v_\delta(s)-M)_+|^{q-2}(v_\delta(s)-M)_+$ and integrate over $\mathbb{R}^N$, we obtain

The first term on the right hand side of (105) is estimated by

Since $v_\delta(s)>M>0$ for $s\in[\tau-1,\tau]$, we have $u_\delta(s) = e^{\int_{0}^{s}\mathcal{G}_\delta(\vartheta_{r-\tau}\omega)dr}v_\delta(s)>0$. Therefore utilizing (7) the nonlinearity term in (105) is bounded by

from which it follows that for each $s\in[\tau-1,\tau]$,

If $v(s)\equiv M$ for $s\in[\tau-1,\tau]$, then (107) also holds true. For the non-autonomous term, using Young's inequality we have

Therefore (107)-(108) together show that

By Lemma 3.2, there exists a $\delta_1 = \delta_1(\tau,\omega)>0$ such that for all $0<|\delta|\leq \delta_1$ and $s\in[\tau-1,\tau]$,

Then (109) is rewrote as

where $\chi(M) = \frac{\alpha_1}{2}e^{-(q-2)C_{\tau,\omega}}M^{q-2}$ and constant $c>0$. Clearly $\lim\limits_{M\rightarrow +\infty}\chi(M)\rightarrow \infty$ for any $q>2$. We apply [37,Lemma 4.2] to over interval $[\tau-1,\tau]$ to yield

In terms of (94), there exist constants $\delta_2 = \delta_2(\tau,\omega)>0$, $\widehat{c} = \widehat{c}(\tau,\omega)>0$ and $T_1 = T_1(\tau,\omega)$ such that for all $t\geq T_1$ and $0<|\delta|\leq\delta_2$,

As for the non-autonomous term of the right in (112), for a large positive number $M$ such that $\chi(M)>\lambda$ and taking $\eta\in(0,1)$, we may derive

for $M$ large enough. Indeed, by (10) the first term on the right hand side of (114) converges to zero when $M\rightarrow +\infty$, by $g\in L^2_{loc}(\mathbb{R}, L^2(\mathbb{R}^N))$ and $\eta$ is small enough that the second term is small enough. Analogously, the integral $\int^\tau_{\tau-1}e^{\chi(M)(s-\tau)} \|\psi_1(s)\|_{\frac{q}{2}}^{\frac{q}{2}}ds$ also approaches zero when $M\rightarrow +\infty$. Put

Then from (112)-(114) it follows that there exists $M_1$ large enough such that

Note that $v_\delta-M_1>\frac{v_\delta}{2}$ for $v_\delta>2M_1$. Then we have the set inclusion $\{x\in {\mathbb{R}}^N: v_\delta-M_1>\frac{v_\delta}{2}\}\supset\{x\in {\mathbb{R}}^N: v_\delta\geq 2M_1\}$. So we see from (115) that

from which and $v_\delta(\tau,\tau-t,\vartheta_{-\tau}\omega,v_{\delta,\tau}) = e^{-\int_{-\tau}^{0}\mathcal{G}_\delta(\vartheta_r\omega)dr}u_{\delta} (\tau,\tau-t,\vartheta_{-\tau}\omega, u_{\delta,\tau})$ we have

By (110), we get that $u_\delta(\tau)\geq2e^{\int_{-\tau}^{0}\mathcal{G}_\delta(\vartheta_r\omega)dr}M_1$ if $u_\delta(\tau)\geq2e^{C_{\tau,\omega}}M_1$. Put $\widetilde{M_1} = 2e^{C_{\tau,\omega}}M_1$. Then it follows from (116) that

Multiplying (26) with $|(v_\delta+M)_-|^{q-2}(v_\delta+M)_-$ and integrate over $\mathbb{R}^N$, where $(v_\delta+M)_-$ is negative part of $v_\delta+M$ for some $M = M(\tau,\omega)>0$, and exerting a similar procedure, we can deduce that there exist positive constants $T_2 = T_2(\tau,\omega), \delta^\prime_{0} = \delta^\prime_{0}(\tau,\omega)$ and $\widetilde{M_2} = \widetilde{M_2}(\tau,\omega)$ such that

Then (117) and (118) together imply the desired.

5.2.   Wong-Zakai approximations in $L^q$

By the result in [19,Theorem 3.1], we know that for every $\delta\neq0$, the $\mathfrak{D}$-pullback random attractor $\mathcal{A}_\delta$ established in $L^2( {\mathbb{R}}^N)$ is also a unique tempered $\mathfrak{D}$-pullback random attractor in $L^q( {\mathbb{R}}^N)$ with $q\geq p>2$, that is to say, for every $\tau \in \mathbb{R}$ and $\omega \in \Omega$, $\mathcal{A}_\delta(\tau, \omega)$ is compact and attracting in the topology of $L^q$.

Proposition 1. Suppose that (7)-(11) and (93) hold and ${\mathfrak{D}}$ is defined by (12). Then for every $\delta\neq0$, the tempered $\mathfrak{D}$-pullback random attractor $\mathcal{A}_\delta$ derived in Theorem 4.7 for the continuous cocycle $\varphi_\delta$ defined by the (28) is also a unique tempered $\mathfrak{D}$-pullback random attractor in $L^q( {\mathbb{R}}^N)$ with $q\geq p>2$. Moreover, for each $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $K_\delta = \{K_\delta(\tau,\omega): \tau\in {\mathbb{R}},\omega\in\Omega)\}$ is given by (50).

We now consider the high-order Wong-Zakai approximation of equation (5) in $L^q( {\mathbb{R}}^N)$. We first show the pre-compactness of the family of attractors $\mathcal{A}_\delta$ in $L^q( {\mathbb{R}}^N)$ as $\delta\rightarrow0$.

Lemma 5.3. Suppose that (7)-(10) and (93) hold. Then for every $\tau \in \mathbb{R}$ and $\omega \in \Omega,$ if $\delta_{n} \rightarrow 0$ as $n\rightarrow \infty$ and $u_{n} \in \mathcal{A}_{\delta_{n}}(\tau, \omega),$ then $\{u_{n}\}_{n = 1}^{\infty}$ has a convergent subsequence in $L^{q}(\mathbb{R}^{N})$.

Proof. According to the invariance of $\mathcal{A}_\delta$ and the decomposability of space ${\mathbb{R}}^N$, along with Lemma 4.10, the proof is quite similar to that of [39,Lemma 5.4]. The details are omitted here.

We are ready to state the main result of this section in the following theorem, which shows the convergence in $L^q( {\mathbb{R}}^N)$ for the tempered random attractors $\mathcal{A}_\delta$ and $\mathcal{A}$ for Eqs.(5) and (1).

Theorem 5.4. Suppose that (7)-(11) and (93) hold. Then for every fixed $\tau \in \mathbb{R}$ and $\omega \in \Omega$.

where $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ and $\mathcal{A}_\delta = \{\mathcal{A}_\delta(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ are the tempered pullback random attractors corresponding to Eqs.(1) and (5), respectively.

Proof. By Theorem 4.9 we have that for all $\tau\in {\mathbb{R}}, t\geq 0$ and $\omega\in\Omega$,

in $L^2(\mathbb{R}^N)$ whenever $\delta_n \rightarrow0$ and $u_{n,\tau}\rightarrow u_{\tau}$ in $L^2(\mathbb{R}^N)$. By Lemma 4.5 it follows that $\limsup_{\delta\rightarrow 0}\|K_{\delta}(\tau, \omega)\|\leq\varrho(\tau, \omega)$. Then in conjunction with Lemma 4.10 and Lemma 5.3 we see that all conditions in Theorem 2.7 are fulfilled, which concludes the proof.

6.   $L^l$-Wong-Zakai approximation $(\forall l> q)$

In this section, we consider the Wong-Zakai approximation in $L^l( {\mathbb{R}}^N)$ for arbitrary $l>q$, which is an interesting work and not done in any literature. To do this, we need to further assume that

By $g\in L_{loc}^2( {\mathbb{R}}; L^2( {\mathbb{R}}^N))$ and $\psi_1\in L_{loc}^\frac{q}{2}( {\mathbb{R}}; L^{\frac{q}{2}}( {\mathbb{R}}^N))$, we know that

for any $l> q$, under which we can prove that the solution is bounded in $L^l( {\mathbb{R}}^N)$ for arbitrary $l>q$. This is done by the traditional mathematical induction technique.

Lemma 6.1. Let $l> q$ be arbitrary. Suppose that (7)-(10) and (119) hold and ${\mathfrak{D}}$ is defined by (12). Given $\tau\in\mathbb{R}, \omega\in\Omega$ and $k\in {\mathbb{N}}$, then for every $u_{\delta,\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ with $D\in {\mathfrak{D}}$, there exist positive constants $\delta_0 = \delta_0(\tau,\omega)$, $c = c(\tau,\omega,l)$ and $T = T(\tau,\omega,l)\geq2$ such that the solution of Eq.(5) satisfies that

Proof. We first prove by induction that there exist positive constants $\delta_0 = \delta_0(\tau,\omega)$, $\widetilde{c}_i^{(k)} = \widetilde{c}_i^{(k)}(\tau,\omega)(i = 1,2)$ and $T_k = T_k(\tau,\omega)\geq2$ such that the solution $v_\delta$ of Eq.(26) satisfies

and

where

If $k = 1$, the result is given in Lemma 5.1. We assume that (120) and (121) hold for $k$, we then prove that it is true for $k+1$.

Multiplying (26) by $|v_\delta|^{qa_{k+1}-2}v_\delta$ and integrating over ${\mathbb{R}}^N$, we have

where we have

For the nonlinearity in (122) by using (7) and Young's inequality we obtain

And for the last term on the right hand side of (122), by Young's inequality again we get

By (122)-(125), we deduce that

We apply [36,Lemma 6.1] in (126) over the interval $[\tau-\frac{1}{k+1},\tau]$ to find that, for every $\xi\in[\tau-\frac{1}{k+1},\tau]$,

By Lemma 3.2, there exists $\delta_1 = \delta_1(\tau,\omega)$ such that for all $0<|\delta|\leq\delta_1$, we have

for all $s\in[-\frac{1}{k},0]$. Then by (127) and our induction assumption (121), we get that there exist positive constants $\delta_2 = \delta_2(\tau,\omega)\leq \delta_1$, $\widetilde{c}_i^{(k)} = \widetilde{c}_i^{(k)}(\tau,\omega)(i = 1,2)$ and $T_k = T_k(\tau,\omega)\geq2$ such that for every $0<|\delta|\leq\delta_2$, $t\geq T_k$ and $\xi\in[\tau-\frac{1}{k+1},\tau]$,

By Lemma 3.2 again, for every $s\in[\tau-\frac{1}{k+1},\tau]$, there exist constants $C_{\tau,\omega}>0$ and $\delta_3 = \delta_3(\tau,\omega)\leq \delta_2$ such that $(q-2)\int_{-\tau}^{s-\tau}\mathcal{G}_\delta(\vartheta_r\omega)dr\geq -(q-2)(C_{\tau,\omega}-\frac{\lambda}{6}(s-\tau))$, then we have

Thus $e^{\frac{4\lambda}{3}(s-\tau)+(q-2)\int_{-\tau}^{s-\tau}\mathcal{G}_\delta(\vartheta_r\omega)dr}\geq e^{-(q-2)C_{\tau,\omega}+(\frac{4}{3}+\frac{q-2}{6})\frac{\lambda}{k+1}}$ for every $0<|\delta|\leq\delta_3$. Therefore by (128), we deduce that there exist positive constants $\widetilde{c}_i^{(k+1)}(\tau,\omega)(i = 1,2)$ such that for every $0<|\delta|\leq\delta_3$ and $t\geq T_k$,

and

For arbitrary $l>q$, there is a $k_0\in {\mathbb{N}}$ such that $q<l< qa_{k_0}$, then by the Sobolev interpolation theorem, there exists $\theta\in(0,1)$ with $\frac{1}{l} = \frac{\theta}{q}+\frac{1-\theta}{qa_{k_0}}$ such that

which along with (94) and (129) (with $\xi = \tau$), we conclude the proof.

By the bound in $L^l( {\mathbb{R}}^N)$ and the Sobolev interpolation, we can obtain the truncation estimate of solution of Eq.(1) in $L^l( {\mathbb{R}}^N)$ for arbitrary $l>q$.

Lemma 6.2. Suppose that (7)-(10) and (119) hold and ${\mathfrak{D}}$ is defined by (12). Let $\tau \in \mathbb{R}$, $\omega \in \Omega$ and $u_{\tau-t}\in D(\tau-t,\vartheta_{-t}\omega)$ with $D\in {\mathfrak{D}}$. Then for any $\varepsilon>0$, there exist positive constants $\delta_0 = \delta_0(\tau,\omega,\varepsilon), c = c(\tau,\omega,l), M_0 = M_0(\tau,\omega,\varepsilon,l)$ and $T = T(\tau,\omega, \varepsilon,l)$ such that the solution $u_\delta$ of Eq.(5) satisfies that

where $(|u_\delta(\tau)|\geq M_0) = \{x\in\mathbb{\mathbb{R}^N}: |u_\delta(\tau,\tau-t,\vartheta_{-\tau}\omega,{u}_{\delta,\tau-t})|\geq M_0\}$.

Proof. By Lemma 6.1, there exist positive constants ${k_0}$, $\delta_1 = \delta_1(\tau,\omega)$, ${c}^{(k_0)} = {c}^{(k_0)}(\tau,\omega)$ and $T_{k_0} = T_{k_0}(\tau,\omega)\geq2$ such that the solution of Eq.(1) satisfies that

By Lemma 5.2, for any $\varepsilon>0$ there exist positive constants $\delta_2 = \delta_2(\tau,\omega,\varepsilon)\geq\delta_0, c = c(\tau,\omega), M_0 = M_0(\tau,\omega,\varepsilon)$ and $T_0 = T_0(\tau,\omega, \varepsilon)$ such that the solution $u_\delta$ of Eq.(1) satisfies that

For arbitrary $l> q$, there exists a $k_0$ such that $q< l< qa_{k_0}$, and then by the Sobolev interpolation we get

where $\theta$ satisfies $\frac{1}{l} = \frac{\theta}{q}+\frac{1-\theta}{a_{k_0}}$. Let $T = \max\{T_0,T_{k_0}\}$ and $\delta_0 = \min\{\delta_1,\delta_2\}$. Then by (131)-(133), it follows that for all $t\geq T$ and $0<|\delta|\leq\delta_0$,

This complete the proof.

We now present the following compactness of random attractor $\mathcal{A}_\delta$ in $L^l( {\mathbb{R}}^N)$ as $\delta\rightarrow0$ for arbitrary $l>q$.

Lemma 6.3. Suppose that (7)-(10), (93) and (119) hold. Then for every $\tau \in \mathbb{R}$ and $\omega \in \Omega,$ if $\delta_{n} \rightarrow 0$ as $n\rightarrow \infty$ and $u_{n} \in \mathcal{A}_{\delta_{n}}(\tau, \omega),$ then $\{u_{n}\}_{n = 1}^{\infty}$ has a convergent subsequence in $L^{l}(\mathbb{R}^{N})$ for arbitrary $l>q$.

Proof. This is followed by a same procedure as in [39,Lemma 5.4], using the asymptotical compactness in $L^q( {\mathbb{R}}^N)$ as in Lemma 5.3. The detailed is omitted here.

We now obtain the upper semi-continuity of random attractors in $L^l( {\mathbb{R}}^N)$ for any $l>q$.

Theorem 6.4. Suppose that (7)-(11), (93) and (119) hold. Let $l>q$ be arbitrary. Then for every fixed $\tau \in \mathbb{R}$ and $\omega \in \Omega$,

where $\mathcal{A} = \{\mathcal{A}(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ and $\mathcal{A}_\delta = \{\mathcal{A}_\delta(\tau,\omega):\tau\in {\mathbb{R}},\omega\in\Omega\}$ are the tempered pullback random attractors corresponding to Eqs.(1) and (5), respectively.

Proof. This is followed by Theorem 4.9, Lemmas 4.5, 4.10, 6.3 and Theorem 2.7.

Acknowledgments

We would like to thank the referees and editors for your valued comments and hard work on our manuscript.