Limiting dynamics for stochastic nonclassical diffusion equations

TL;DR

This paper investigates the limiting behavior of stochastic nonclassical diffusion equations, showing that as certain parameters tend to zero, solutions converge to those of stochastic heat equations, with results valid for cubic nonlinearities.

Contribution

It establishes the inviscid limits of solutions to stochastic nonclassical diffusion equations, including convergence in probability, using uniform estimates and regularity theory.

Findings

Inviscid limits reduce to stochastic heat equations.

Convergence in probability in L^2(0,T;H^1) is proved.

Results hold for cubic nonlinearities.

Abstract

In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic heat equations. We deal with initial values in $H^{1}_{0}(I)$ and $H^{2}(I)\cap H^{1}_{0}(I)$. When the initial value in $H^{1}_{0}(I),$ we establish the inviscid limits of the weak martingale solution; when the initial value in $H^{2}(I)\cap H^{1}_{0}(I),$ we establish the inviscid limits of the weak solution, the convergence in probability in $L^{2}(0,T;H^{1}(I))$ is proved.The results are valid for cubic nonlinearity. The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.

Equations

$$u_t-\Delta u_t-\Delta u+u^3-u=0$$

$$\begin{array}[]{l}\left\{\begin{array}[]{llll}d(u^{\varepsilon}-\varepsilon u^{\varepsilon}_{xx})+(-u^{\varepsilon}_{xx}+u^{\varepsilon 3}-u^{\varepsilon})dt=g(u^{\varepsilon})dB\\ u^{\varepsilon}(0,t)=0=u^{\varepsilon}(1,t)\\ u^{\varepsilon}(0)=u_{0}\par\end{array}\right.\end{array}\begin{array}[]{lll}{\rm{in}}~{}I\times(0,T)\\ {\rm{in}}~{}(0,T)\\ {\rm{in}}~{}I,\end{array}$$

$$u_t-\epsilon \Delta u_t-\Delta u+u^3-u=0,$$

$$\begin{array}[]{l}\begin{array}[]{llll}\|g(u)\|_{L^{2}(I)}\leq L(1+\|u\|_{L^{2}(I)})~{}~{}\forall u\in L^{2}(I),\\ \|g(u_{1})-g(u_{2})\|_{L^{2}(I)}\leq L\|u_{1}-u_{2}\|_{L^{2}(I)}~{}~{}\forall u_{1},u_{2}\in L^{2}(I).\end{array}\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}\|g(u)\|_{L^{2}(I)}\leq L(1+\|u\|_{L^{2}(I)})~{}~{}\forall u\in L^{2}(I),\\ \|g(u)\|_{H^{1}(I)}\leq L(1+\|u\|_{H^{1}(I)})~{}~{}\forall u\in H^{1}(I),\\ \|g(u_{1})-g(u_{2})\|_{H^{1}(I)}\leq L\|u_{1}-u_{2}\|_{H^{1}(I)}~{}~{}\forall u_{1},u_{2}\in H^{1}(I)\par.\end{array}\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}[(u(t),\varphi)+\varepsilon(u_{x}(t),\varphi_{x})]-[(u_{0},\varphi)+\varepsilon(u_{0x},\varphi_{x})]+\displaystyle\int_{0}^{t}((u_{x},\varphi_{x})+(u^{3}-u,\varphi))ds\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\displaystyle\int_{0}^{t}(g(u),\varphi)dB\end{array}\end{array}$$

$$\mathbb{E}\underset{0\le t\le T}{\sup }(\Vert u^{\epsilon }(t){\Vert }_{L^2(I)}^2+\epsilon \Vert u_x^{\epsilon }(t){\Vert }_{L^2(I)}^2{)}^{\frac{p}{2}}\le C(p,T),$$

$$\mathbb{E}{\left({\int }_0^T(\Vert u_x^{\epsilon }(t){\Vert }_{L^2(I)}^2+\Vert u^{\epsilon }{\Vert }_{L^4(I)}^4)dt\right)}^{\frac{p}{2}}\le C(p,T),$$

$$\mathbb{E}\underset{0\le |\theta |\le \delta \le 1}{\sup }{\int }_0^T\Vert u^{\epsilon }(t+\theta )-u^{\epsilon }(t){\Vert }_{H^{-1}(I)}^{2}dt\le C(p,T)\delta ,$$

$$u_1=u_{2}P-a.s.\mathrm{for}\mathrm{all}t\in [0,T].$$

$$\begin{array}[]{l}\begin{array}[]{llll}\|g(t,u)\|_{L^{2}(I)}\leq C(1+\|u\|_{L^{2}(I)})~{}~{}\forall t\in[0,T]~{}~{}\forall u\in L^{2}(I),\\ \|g(t,u_{1})-g(t,u_{2})\|_{L^{2}(I)}\leq C\|u_{1}-u_{2}\|_{L^{2}(I)}~{}~{}\forall u_{1},u_{2}\in L^{2}(I),\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\mathcal{L}(\tilde{u}^{\varepsilon_{i}},\tilde{B}^{\varepsilon_{i}})=\mathcal{L}(u^{\varepsilon_{i}},B^{\varepsilon_{i}})\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\tilde{u}^{\varepsilon_{i}}\rightarrow u~{}~{}{\rm{strongly~{}~{}in}}~{}~{}L^{2}(\Omega,L^{2}(0,T;L^{2}(I))),\\ \tilde{u}^{\varepsilon_{i}}\rightarrow u~{}~{}{\rm{weakly~{}~{}in}}~{}~{}L^{p}(\Omega,L^{2}(0,T;H^{1}(I))),\\ \tilde{u}^{\varepsilon_{i}}\rightarrow u~{}~{}{\rm{weakly~{}~{}star~{}~{}in}}~{}~{}L^{p}(\Omega,L^{\infty}(0,T;L^{2}(I))),\\ \tilde{B}^{\varepsilon_{i}}\rightarrow B~{}~{}{\rm{in}}~{}~{}C([0,T];\mathbb{R}^{1})~{}~{}\mathbb{P}-a.s.,\par\par\end{array}\end{array}$$

$$\begin{array}[]{l}\left\{\begin{array}[]{llll}du+(-u_{xx}+u^{3}-u)dt=g(u)dB\\ u(0,t)=0=u(1,t)\\ u(0)=u_{0}\par\end{array}\right.\end{array}\begin{array}[]{lll}{\rm{in}}~{}I\times(0,T)\\ {\rm{in}}~{}(0,T)\\ {\rm{in}}~{}I.\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}\|g(t,u)\|_{L^{2}(I)}\leq C(1+\|u\|_{L^{2}(I)})~{}~{}\forall t\in[0,T]~{}~{}\forall u\in L^{2}(I),\\ \|g(t,u_{1})-g(t,u_{2})\|_{L^{2}(I)}\leq C\|u_{1}-u_{2}\|_{L^{2}(I)}~{}~{}\forall u_{1},u_{2}\in L^{2}(I),\end{array}\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}(u(t),\varphi)-\varepsilon(u(t),\varphi_{xx})\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=(u_{0},\varphi)-\varepsilon(u_{0},\varphi_{xx})+\displaystyle\int_{0}^{t}(u(s),\varphi_{xx})ds-\displaystyle\int_{0}^{t}(u^{3}-u,\varphi)ds+\displaystyle\int_{0}^{t}(g(s),\varphi)dB(s)\end{array}\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}\|u^{\varepsilon}(t)\|_{L^{2}(I)}^{2p}+\mathbb{E}(\displaystyle\int_{0}^{T}\|u^{\varepsilon}_{x}\|_{L^{2}(I)}^{2}dt)^{p}+\mathbb{E}(\displaystyle\int_{0}^{T}\displaystyle\int_{I}u^{\varepsilon 4}dxdt)^{p}+\mathbb{E}(\displaystyle\int_{0}^{T}\varepsilon\|u^{\varepsilon}_{xx}\|_{L^{2}(I)}^{2}dt)^{p}\\ \leq C(p,L,T,I,u_{0}).\end{array}\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}(\|u^{\varepsilon}_{x}(t)\|_{L^{2}(I)}^{2}+\varepsilon\|u^{\varepsilon}_{xx}(t)\|_{L^{2}(I)}^{2})+\mathbb{E}\displaystyle\int_{0}^{T}\|u^{\varepsilon}_{xx}\|_{L^{2}(I)}^{2}dt\leq C(L,T,I,u_{0}).\par\end{array}\end{array}$$

$$\begin{array}[]{l}\left\{\begin{array}[]{llll}dz+(-z_{xx}+z^{3}-z)dt=g(z)dB\\ z(0,t)=0=z(1,t)\\ z(0)=u_{0}\par\end{array}\right.\end{array}\begin{array}[]{lll}{\rm{in}}~{}I\times(0,T)\\ {\rm{in}}~{}(0,T)\\ {\rm{in}}~{}I,\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\lim\limits_{\varepsilon\rightarrow 0}\mathbb{P}(\|u^{\varepsilon}-z\|_{L^{2}(0,T;H^{1}(I))}>\delta)=0.\par\end{array}\end{array}$$

$$\underset{\theta \to 0}{\lim }{\int }_0^{T-\theta }\Vert v(t+\theta )-v(t){\Vert }_Y^{p}dt=0,$$

$$\displaystyle\begin{array}[]{l}\mathcal{Z}=\left\{q\in\left|\begin{array}[]{llll}L^{2}(0,T;X)\displaystyle\cap L^{\infty}(0,T;B)\\ \displaystyle\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\left(\displaystyle\int_{0}^{T}\|q(t+\theta)-q(t)\|_{Y}^{2}dt\right)^{\frac{1}{2}}<+\infty\end{array}\right.\right\}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\|y\|_{Y_{\mu_{m},\nu_{m}}^{1}}=\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{L^{2}(I)}+\left(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{1}(I)}^{2}dt\right)^{\frac{1}{2}}\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\displaystyle+\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{H^{-1}(I)}^{2}dt.\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{L^{2}(I)}^{2p}<\infty,~{}~{}~{}\mathbb{E}\left(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{1}(I)}^{2}dt\right)^{\frac{p}{2}}<\infty,\\ \displaystyle\mathbb{E}\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\left(\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{H^{-1}(I)}^{2}dt\right)^{\frac{1}{2}}<\infty,\par\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\|y\|_{X_{p,\mu_{m},\nu_{m}}^{1}}=\left(\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{L^{2}(I)}^{2p}\right)^{\frac{1}{2p}}+\left(\mathbb{E}(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{1}(I)}^{2}dt)^{\frac{p}{2}}\right)^{\frac{2}{p}}\\ \displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mathbb{E}\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\left(\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{H^{-1}(I)}^{2}dt\right)^{\frac{1}{2}},\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\|y\|_{Y_{\mu_{m},\nu_{m}}^{2}}=\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{H^{1}(I)}+\left(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{2}(I)}^{2}dt\right)^{\frac{1}{2}}\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\displaystyle+\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{L^{2}(I)}^{2}dt.\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{H^{1}(I)}^{2p}<\infty,~{}~{}~{}\mathbb{E}\left(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{2}(I)}^{2}dt\right)^{\frac{p}{2}}<\infty,\\ \displaystyle\mathbb{E}\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\left(\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{L^{2}(I)}^{2}dt\right)^{\frac{1}{2}}<\infty,\par\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{l}\begin{array}[]{llll}\displaystyle\|y\|_{X_{p,\mu_{m},\nu_{m}}^{2}}=\left(\mathbb{E}\displaystyle\sup\limits_{0\leq t\leq T}\|y(t)\|_{H^{1}(I)}^{2p}\right)^{\frac{1}{2p}}+\left(\mathbb{E}(\displaystyle\int_{0}^{T}\|y(t)\|_{H^{2}(I)}^{2}dt)^{\frac{p}{2}}\right)^{\frac{2}{p}}\\ \displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\mathbb{E}\displaystyle\sup\limits_{m}\frac{1}{\nu_{m}}\left(\displaystyle\sup\limits_{|\theta|\leq\mu_{m}}\displaystyle\int_{0}^{T-\theta}\|y(t+\theta)-y(t)\|_{L^{2}(I)}^{2}dt\right)^{\frac{1}{2}},\end{array}\end{array}$$

$$\mathcal{L}(\mathcal{A})=P\{\omega \in \Omega :{\xi }_n(\omega )\in \mathcal{A}\},$$

$$\begin{array}[]{l}\left\{\begin{array}[]{llll}d(u-\varepsilon\Delta u)+(-\Delta u+f)dt=gdB\\ u(x,t)=0\\ u(0)=u_{0}\par\end{array}\right.\end{array}\begin{array}[]{lll}{\rm{in}}~{}G\times(0,T)\\ {\rm{in}}~{}\partial G\times(0,T)\\ {\rm{in}}~{}G,\end{array}$$

$$\begin{array}[]{l}\begin{array}[]{llll}(u(t),\varphi)-\varepsilon(u(t),\triangle\varphi)\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=(u_{0},\varphi)-\varepsilon(u_{0},\triangle\varphi)+\displaystyle\int_{0}^{t}(u(s),\triangle\varphi)ds-\displaystyle\int_{0}^{t}(f(s),\varphi)ds+\displaystyle\int_{0}^{t}(g(s),\varphi)dB(s)\end{array}\end{array}$$

Full text

The key points in the proof of our convergence results are establishing some uniform estimates and the regularity theory for the solutions of the stochastic nonclassical diffusion equations which are independent of the parameter. Based on the uniform estimates, the tightness of distributions of the solutions can be obtained.

Keywords: Inviscid limits; Singular perturbation; Stochastic nonclassical diffusion equation; Stochastic heat equation; Weak martingale solution; Weak solution; Tightness

2010 Mathematics Subject Classification: 60H15, 35K70, 35Q35, 35A01

1 Introduction

Nonclassical parabolic equation

[TABLE]

arises as a model to describe physical phenomena such as non-Newtonian flow, soil mechanics and heat conduction, etc.; see [1, 5, 24, 33, 34] and references therein. Aifantis [1] provides a quite general approach for obtaining these equations.

In a number of applications, the systems are subject to stochastic fluctuations arising as a result of either uncertain forcing (stochastic external forcing) or uncertainty of the governing laws of the system. The need for taking random effects into account in modeling, analyzing, simulating and predicting complex phenomena has been widely recognized in geophysical and climate dynamics, materials science, chemistry, biology and other areas. Stochastic partial differential equations (SPDEs or stochastic PDEs) are appropriate mathematical models for complex systems under random influences [37]. The fact that in physical experiments there are always small irregularities which give birth to a new random phenomenon justifies the study of equations with noise.

In this paper, we investigate

[TABLE]

where $\varepsilon\in[0,1),I=[0,1],T>0.$ This paper is concerned with the asymptotic behavior of solutions of (1.1) as $\varepsilon\rightarrow 0.$

For the deterministic nonclassical diffusion equation

[TABLE]

[35] establishs some uniform decay estimates for the solutions which are independent of the parameter $\varepsilon$, then they prove the continuity of solutions as $\varepsilon\rightarrow 0.$ Upper semicontinuity of the family of global attractors at $\varepsilon=0$ in the topology of $H^{1}_{0}$ is also established. [2] considers the first initial boundary value problem for the non-autonomous nonclassical diffusion equation. By using the asymptotic a priori estimate method, the authors prove the existence of pullback attractors and the upper semicontinuity of pullback attractors.

For the stochastic nonclassical diffusion equations, [38] concerns the dynamics of this equation on $\mathbb{R}^{N}$ perturbed by a $\varepsilon$-random term. By using an energy approach, the authors prove the asymptotic compactness of the associated random dynamical system, and then the existence of random attractors. Finally, they show the upper semicontinuity of random attractors in the sense of Hausdorff semi-metric. [3, 39] prove the existence of pullback attractor for stochastic nonclassical diffusion equations on unbounded domains with non-autonomous deterministic and stochastic forcing terms, and by using a tail-estimates method, the authors establish the pullback asymptotic compactness of the random dynamical system.

In recent years, many efforts have been devoted to studying the singularly perturbed nonlinear SPDEs.

[6, 7, 8, 9, 10] consider the Smoluchowski-Kramers approximation the singularly perturbed nonlinear stochastic wave equations. In [18] relations between the asymptotic behavior for a stochastic wave equation and a heat equation are considered. The upper semicontinuity of global random attractor and the global attractor of the heat equation is investigated. Furthermore they shows that the stationary solutions of the stochastic wave equation converge in probability to some stationary solution of the heat equation. [36] studies a continuity property for the measure attractors of the singularly perturbed nonlinear stochastic wave equations, any one stationary solution of the limit heat equation is a limit point of a stationary solution of the singularly perturbed nonlinear stochastic wave equations. An averaging method is applied to derive effective approximation to a singularly perturbed nonlinear stochastic damped wave equation in [19]. [20] establishes a large deviation principle for the singularly perturbed stochastic nonlinear damped wave equations. In [21], the random inertial manifold of a stochastic damped nonlinear wave equations with singular perturbation is proved to be approximated almost surely by that of a stochastic nonlinear heat equation which is driven by a new Wiener process depending on the singular perturbation parameter.

[28] establishs the weak martingale solution for stochastic model for two-dimensional second grade fluids and studied their behaviour when $\alpha\rightarrow 0$. [13] studies the asymptotic behavior of weak solutions to the stochastic 3D Navier-Stokes-$\alpha$ model as $\alpha\rightarrow 0$, the main result provides a new construction of the weak solutions of stochastic 3D Navier-Stokes equations as approximations by sequences of solutions of the stochastic 3D Navier-Stokes-$\alpha$ model. [32] discusses the relation of the stochastic 3D magnetohydrodynamic-$\alpha$ model to the stochastic 3D magnetohydrodynamic equations by proving a convergence theorem, that is, as the length scale $\alpha\rightarrow 0$, a subsequence of weak martingale solutions of the stochastic 3D magnetohydrodynamic-$\alpha$ model converges to a certain weak martingale solution of the stochastic 3D magnetohydrodynamic equations.

However, there are very few results for the limiting dynamics for stochastic nonclassical diffusion equations with singularly perturbed.

Motivated by previous research and from both physical and mathematical standpoints, the following mathematical questions arise naturally which are important from the point of view of dynamical systems:

• •

Does the solution $u^{\varepsilon}$ for (1.1) converge as $\varepsilon\rightarrow 0$?

• •

If $u^{\varepsilon}$ converges as $\varepsilon\rightarrow 0$, what is the limit of $u^{\varepsilon}$?

In this paper we will answer the above problems. The question of asymptotic analysis of partial differential equations when some physical parameters converge to some limit has always been of great interest.

To the best of our knowledge, it is the first contribution to the literature on this problem.

Through this paper, we make the following assumptions:

H1) Let $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\geq 0},P)$ be a complete filtered probability space on which a one-dimensional standard Brownian motion $\{B(t)\}_{t\geq 0}$ is defined such that $\{\mathcal{F}_{t}\}_{t\geq 0}$ is the natural filtration generated by $w(\cdot),$ augmented by all the $P-$ null sets in $\mathcal{F}.$ Let $H$ be a Banach space, and let $C([0,T];H)$ be the Banach space of all $H-$valued strongly continuous functions defined on $[0,T].$ We denote by $L_{\mathcal{F}}^{p}(0,T;H)(1\leq p<+\infty)$ the Banach space consisting of all $H-$valued $\{\mathcal{F}_{t}\}_{t\geq 0}-$adapted processes $X(\cdot)$ such that $E(\|X(\cdot)\|^{p}_{L^{p}(0,T;H)})<\infty;$ by $L_{\mathcal{F}}^{\infty}(0,T;H)$ the Banach space consisting of all $H-$valued ${\mathcal{\{}\mathcal{F}_{t}}\}_{t\geq 0}-$adapted bounded processes; by $L_{\mathcal{F}}^{2}(\Omega;C([0,T];H))$ the Banach space consisting of all $H-$valued $\{\mathcal{F}_{t}\}_{t\geq 0}-$adapted continuous processes $X(\cdot)$ such that $E(\|X(\cdot)\|^{2}_{C([0,T];H)})<\infty.$ All the above spaces are endowed with the canonical norm.

H2) For a random variable $\xi$, we denote by $\mathcal{L}(\xi)$ its distribution.

H3) $(\cdot,\cdot)$ stands for the inner product in $L^{2}(I)$.

H4) The letter $C$ with or without subscripts denotes positive constants whose value may change in different occasions. We will write the dependence of constant on parameters explicitly if it is essential.

We make the the two different assumptions on $g.$

(A) $g\in C(\mathbb{R})$ and there exists a constant $L>0$ such that

[TABLE]

(B) $g\in C(\mathbb{R})$ and there exists a constant $L>0$ such that

[TABLE]

1.1 Weak martingale solution

Definition 1.1**.**

A weak martingale solution of (1.1) is a system $\{(\Omega,\mathcal{F},\mathbb{P}),$ $(\mathcal{F}_{t})_{0\leq t\leq T},u,B\},$ where

(1) $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space,

(2) $(\mathcal{F}_{t})_{0\leq t\leq T}$ is a filtration satisfying the usual condition on $(\Omega,\mathcal{F},\mathbb{P}),$

(3) $B$ is a $\mathcal{F}_{t}-$adapted $\mathbb{R}-$valued Wiener process,

(4) $u\in L^{p}(\Omega,L^{\infty}(0,T;L^{2}(I)))\displaystyle\cap L^{p}(\Omega,L^{2}(0,T;H^{1}(I)))\displaystyle\cap L^{2p}(\Omega,L^{4}(0,T;L^{4}(I))),$ for every $1\leq p\leq\infty,$

(5) For all $\varphi\in H^{1}_{0}(I),$

[TABLE]

hold $dt\otimes d\mathbb{P}-$almost everywhere.

(6) The function $u(t)$ take values in $L^{2}(I)$ and is continuous with respect to $t$ $\mathbb{P}-$almost surely.

The first main result of this paper is given in the next statement.

Theorem 1.1**.**

Let assumption (A) be satisfied, $T>0$ and $u_{0}\in H^{1}_{0}(I).$ For any $\varepsilon\in[0,\frac{1}{2}],$ there exists a weak martingale solution $\{(\Omega^{\varepsilon},\mathcal{F}^{\varepsilon},\mathbb{P}^{\varepsilon}),(\mathcal{F}^{\varepsilon}_{t})_{0\leq t\leq T},u^{\varepsilon},B^{\varepsilon}\}$ of problem (1.1) such that the following estimates hold for any $1\leq p<\infty:$

[TABLE]

where $C(p,T)$ is a constant independent of $\varepsilon.$

Moreover, let $u_{1}$ and $u_{2}$ be two weak martingale solutions of problem (1.1) defined on the same prescribed stochastic basis $\{(\Omega,\mathcal{F},P),(\mathcal{F}_{t})_{0\leq t\leq T},B\}$ starting with the same initial condition $u_{0},$ then

[TABLE]

Remark 1.1**.**

If we replace $g(u)$ in (1.1) by $g(t,u)$ and assume that $g(t,u)$ is nonlinear measurable mapping defined on $[0,T]\times L^{2}(I)$ taking values on $L^{2}(I),$ it is continuous with respect to $u$ and there exists a constant $C$ such that

[TABLE]

the conclusion in Theorem 1.1 also holds.

Remark 1.2**.**

Theorem 1.1 is established by the compactness method combines the Galerkin approximation scheme with sharp compactness results in function spaces of Sobolev type due to Simon and some celebrated probabilistic compactness results of Prokhorov and Skorokhod.

Asymptotic behavior of the weak martingale solutions for the stochastic nonclassical diffusion equations as $\varepsilon\rightarrow 0$ can be described by the following results.

Theorem 1.2**.**

Let assumption (A) be satisfied, $T>0$ and $u_{0}\in H^{1}_{0}(I).$ If $\{(\Omega^{\varepsilon},\mathcal{F}^{\varepsilon},\mathbb{P}^{\varepsilon}),(\mathcal{F}^{\varepsilon}_{t})_{0\leq t\leq T},u^{\varepsilon},B^{\varepsilon}\}_{\varepsilon\in[0,1]}$ are the weak martingale solutions of problem (1.1), there exists a subsequence $\{\varepsilon_{i}\}\subset[0,1]$ with $\varepsilon_{i}\rightarrow 0$ as $i\rightarrow\infty$, a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and random variables $(\tilde{u}^{\varepsilon_{i}},\tilde{B}^{\varepsilon_{i}}),$ $(u,B)$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in $L^{2}(0,T;L^{2}(I))\times C([0,T];\mathbb{R}^{1})$ such that

[TABLE]

and the following convergences hold for any $1\leq p<\infty:$

[TABLE]

as $i\rightarrow\infty$ and $\{(\Omega,\mathcal{F},\mathbb{P}),(\mathcal{F}_{t})_{0\leq t\leq T},u,B\}$ is a weak martingale solution of problem

[TABLE]

Remark 1.3**.**

If we replace $g(u)$ in (1.1) by $g(t,u)$ and assume that $g(t,u)$ is nonlinear measurable mapping defined on $[0,T]\times L^{2}(I)$ taking values on $L^{2}(I),$ it is continuous with respect to $u$ and there exists a constant $C$ such that

[TABLE]

the conclusion in Theorem 1.2 also holds.

1.2 Weak solution

Next, we consider another kind of solution to (1.1).

Definition 1.2**.**

A stochastic process $u$ is said to be a weak solution of (1.1) if

$u$* is $L^{2}(I)$-valued and $\mathcal{F}_{t}$-measurable for each $t\in[0,T],$*

$u\in L^{2}(\Omega;L^{2}(C([0,T];L^{2}(I))),$**

$u(0)=u_{0}$**

and

[TABLE]

holds for all $t\in[0,T]$ and all $\varphi\in H^{2}(I)\displaystyle\cap H^{1}_{0}(I)$, for almost all $\omega\in\Omega.$

Remark 1.4**.**

The weak solution of SPDEs has been discussed in [12].

Theorem 1.3**.**

Let assumption (B) be satisfied, $T>0$ and $u_{0}\in H^{2}(I)\displaystyle\cap H^{1}_{0}(I).$ For any $\varepsilon\in[0,\frac{1}{2}],$ there exists a unique weak solution $u^{\varepsilon}(t)$ to (1.1) in $L^{2}(\Omega;C([0,T];H^{2}(I)\displaystyle\cap H^{1}_{0}(I)))$ and for any $1\leq p<\infty,$ there exists a constant $C(p,L,T,I,u_{0})$ such that

[TABLE]

Moreover, there exists a constant $C(L,T,I,u_{0})$ such that

[TABLE]

Remark 1.5**.**

Since nonlinear terms $u^{3}-u$ are not Lipschitz continuous, we will use a truncation argument which will lead to a local existence result. Then via some a priori estimates we obtain that the solution is also global.

Asymptotic behavior of the weak solutions for the stochastic nonclassical diffusion equations as $\varepsilon\rightarrow 0$ can be described by the following results.

Theorem 1.4**.**

Let assumption (B) be satisfied, $T>0$ and $u_{0}\in H^{2}(I)\displaystyle\cap H^{1}_{0}(I).$ For any $\varepsilon\in[0,\frac{1}{2}],$ if $u^{\varepsilon}$ is the weak solution to (1.1) and $z$ is the weak solution to

[TABLE]

then $u^{\varepsilon}$ converges in probability to $z$ in $L^{2}(0,T;H^{1}(I))$ as $\varepsilon\rightarrow 0,$ namely, for any $\delta>0,$ we have

[TABLE]

1.3 Main difficulties

The main difficulties in this paper are the following respects:

• •

Multiplicative type noise. The noise in equation (1.1) is not additive type, (1.1) is perturbed by a stochastic term of multiplicative type, thus the method in [35, 38, 39] can not be used in dealing with (1.1), we should take new measure. Here the presence of a diffusion coefficient $g$ in front of the stochastic perturbation which is nonconstant makes the proof of Theorem 1.2 and Theorem 1.4 definitely more delicate and requires some extra work which is not necessary in the case of a Gaussian perturbation.

• •

“BBM” term. Equation (1.1) contains the “BBM” term $-u_{xxt}$, its stochastic from is $-du_{xx}$, this brings us new difficulty in establishing the existence and regularity theory for the stochastic nonclassical diffusion equations. In the present work we will try to overcome this difficulty by developing the Galerkin approximation techniques in [22, 15, 16, 17]. The “BBM” term is different from the usual reaction-diffusion equation essentially. For example, the nonclassical diffusion equation does not have smoothing effect, e.g., if the initial data only belongs to a weaker topology space, the solution can not belong to a stronger topology space with higher regularity. Moreover, since the existence of this term, we can’t use the Itö formula to $u^{2}.$ We borrow an essential idea from [22, 15, 16, 17], but substantial technical adaptation is necessary for the problem in this paper.

• •

Uniform estimates independent of the parameter $\varepsilon$. Since the parameter $\varepsilon$ in singular perturbation problem (1.1) is small, the uniform estimates for the solution of (1.1) which are independent of the parameter $\varepsilon$ are very hard to obtain. The proof of the convergence result requires uniform estimates on the Sobolev regularity in space and in time for the solutions to the stochastic nonclassical diffusion equation. As known, such uniform bounds are used to establish tightness property of $u^{\varepsilon}$ in an appropriate functional space.

• •

The cubic non-linear term. The last difficulty arises from polynomial nonlinearity in equation (1.1), the nonlinear term in (1.1) is cubic term $u^{3}-u$, the main obstacle is that it is difficult to obtain a higher regularity estimate to guarantee the continuous convergence of the solutions as $\varepsilon\rightarrow 0$. This type of nonlinearity can be handled by the truncation method. In order to overcome the problem, we use the cut-off technique and the Gagliardo-Nirenberg inequality.

This paper is organized as follows. In Section 2, we give some preliminaries and gather all the necessary tools. The existence of weak martingale solutions for (1.1) is discussed in Section 3, we introduce a Galerkin approximation scheme for the problem (1.1) and obtain a priori estimates for the approximating solutions, then we prove the crucial result of tightness of Galerkin s solutions and apply Prokhorov s and Skorokhod s compactness results to prove Theorem 1.1. Section 4 is concerned with the continuity of weak martingale solutions for (1.1) as $\varepsilon\rightarrow 0$. We derive the results of the tightness of the corresponding probability measures and perform the passage to the limit which establishes the convergence of weak martingale solutions. In Section 5, applying the Picard iteration method to the corresponding truncated equation, we give the local existence of weak solutions to (1.1). Then, the energy estimate shows that the weak solution is also global in time. Moreover, we obtain the uniform estimates for the solution of (1.1) which are independent of the parameter $\varepsilon$. Section 6 is concerned with the continuity of weak solutions for (1.1) as $\varepsilon\rightarrow 0$. We derive tightness property of weak solutions in $L^{2}(0,T;H^{1}(I))$ and perform the passage to the limit which establishes the convergence of weak solutions.

2 Preliminary

This section is devoted to some preliminaries for the proof of Theorem 1.1–Theorem 1.4.

2.1 Some tools

The following compactness results is important for tightness property of Galerkin solutions.

Lemma 2.1**.**

(See [29, Theorem 5]) Let $X,B$ and $Y$ be some Banach spaces such that $X$ is compactly embedded into $B$ and let $B$ be a subset of $Y.$ For any $1\leq p,q\leq\infty,$ let $V$ be a set bounded in $L^{q}(0,T;X)$ such that

[TABLE]

uniformly for all $v\in V.$ Then $V$ is relatively compact in $L^{p}(0,T;B).$

According to Lemma 2.1, we can obtain the following compactness result.

Corollary 2.1**.**

Let $X,B$ and $Y$ satisfy the same assumptions in Lemma 2.1 and $\mu_{m},\nu_{m}$ be two sequences which converge to zero as $m\rightarrow\infty.$ Then

[TABLE]

in $L^{2}(0,T;B)$ is compact.

Remark 2.1**.**

The above compactness result plays a crucial role in the proof of the tightness of the probability measures generated by the sequence $\{u^{\varepsilon}\}_{\varepsilon>0}.$

Now we introduce several spaces which will be used in the next section. Let $\mu_{m},\nu_{m}$ be two sequences that defined in Corollary 2.1.

$\bullet$ The space $Y_{\mu_{m},\nu_{m}}^{1}$ is a Banach space with the norm

[TABLE]

$X_{p,\mu_{m},\nu_{m}}^{1}$ is a space consist of all random variables $y$ on $(\Omega,\mathcal{F},\mathbb{P})$ which satisfy

[TABLE]

where $\mathbb{E}$ denotes the mathematical expectation with respect to the probability measure $\mathbb{P}$. Endowed with the norm

[TABLE]

$X_{p,\mu_{m},\nu_{m}}^{1}$ is a Banach space.

$\bullet$ The space $Y_{\mu_{m},\nu_{m}}^{2}$ is a Banach space with the norm

[TABLE]

$X_{p,\mu_{m},\nu_{m}}^{2}$ is a space consist of all random variables $y$ on $(\Omega,\mathcal{F},\mathbb{P})$ which satisfy

[TABLE]

where $\mathbb{E}$ denotes the mathematical expectation with respect to the probability measure $\mathbb{P}$. Endowed with the norm

[TABLE]

$X_{p,\mu_{m},\nu_{m}}^{2}$ is a Banach space.

In order to pass from martingale to pathwise solutions we make essential use of an elementary but powerful characterization of convergence in probability as given in [14].

Lemma 2.2**.**

(Gyöngy-Krylov Theorem)(See [14, Lemma 1.1],[26, Proposition 6.3]) Let $E$ be a Polish space equipped with the Borel $\sigma$-algebra. A sequence of $E$-valued random element $z_{n}$ converges in probability if and only if for every pair of subsequences $z_{l},z_{m}$ there exists a subsequence $w_{k}=(z_{l(k)},z_{m(k)})$ converging weakly to a random element $w$ supported on the diagonal $\{(x,y)\in E\times E:x=y\}.$

Prokhorov’s Theorem and Skorohod’s Theorem will be used to establish the tightness of $u^{\varepsilon}.$ The following two lemmas will play crucial roles in the proof of Theorem 1.3.

Lemma 2.3** (Prokhorov’s Theorem).**

A sequence of measures $\{\mu_{n}\}$ on $(E,\mathcal{B}(E))$ is tight if and only if it is relatively compact, that is there exists a subsequence $\{\mu_{n_{k}}\}$ which weakly converges to a probability measure $\mu.$

Lemma 2.4** (Skorohod’s Theorem).**

For an arbitrary sequence of probability measures $\{\mu_{n}\}$ on $(E,\mathcal{B}(E))$ weakly converges to a probability measure $\mu,$ there exists a probability space $(\Omega,\mathcal{F},P)$ and random variables $\xi,\xi_{1},...,\xi_{n},...$ with values in $E$ such that the probability law of $\xi_{n},$

[TABLE]

for all $\mathcal{A}\in\mathcal{F},$ is $\mu_{n},$ the probability law of $\xi$ is $\mu,$ and $\displaystyle\lim\limits_{n\rightarrow\infty}\xi_{n}=\xi,$ $P-a.s.$

2.2 The linear stochastic nonclassical diffusion equations

This section is devoted to some preliminaries for the proof of Theorem 1.3.

In this subsection, we let $G$ be the bounded domain of $\mathbb{R}^{n}(n\geq 1).$ We will use the results in this subsection with $n=1$ in Section 5.

Definition 2.1**.**

A stochastic process $u$ is said to be a solution of

[TABLE]

if

$u$* is $L^{2}(G)$-valued and $\mathcal{F}_{t}$-measurable for each $t\in[0,T],$*

$u\in L^{2}(\Omega;C([0,T];L^{2}(G))),$**

$u(0)=u_{0}$**

and

[TABLE]

holds for all $t\in[0,T]$ and all $\varphi\in H^{2}(G)\displaystyle\cap H^{1}_{0}(G)$, for almost all $\omega\in\Omega.$

Lemma 2.5**.**

(See [27, Theorem 8.94]) There exists a set of positive real numbers $\{\lambda_{k}\}_{k\in N}$ such that the corresponding solutions $\{e_{k}\}_{k\in N}$ of the problem

[TABLE]

form a basis in $H^{2}(G)\displaystyle\cap H^{1}_{0}(G),$ which is orthonormal in $L^{2}(G).$

Proposition 2.1**.**

For any $\varepsilon\in[0,1],$ there exists a constant $C$ independent of $\varepsilon.$

1) If $u_{0}\in L^{2}(\Omega;L^{2}(G)),f\in L^{2}(\Omega;L^{2}(0,T;H^{-1}(G))),g\in L^{2}(\Omega;L^{2}(0,T;L^{2}(G)))$, then (2.15) has a unique solution $u\in L^{2}(\Omega;C([0,T];L^{2}(G)))$ and

[TABLE]

2) If $u_{0}\in L^{2}(\Omega;H^{1}_{0}(G)),f\in L^{2}(\Omega;L^{2}(0,T;H^{-1}(G))),g\in L^{2}(\Omega;L^{2}(0,T;L^{2}(G)))$, then (2.15) has a unique solution $u\in L^{2}(\Omega;C([0,T];H^{1}_{0}(G)))\displaystyle\cap L^{2}(\Omega,L^{2}(0,T;H^{1}(G)))$ and

[TABLE]

Moreover, it holds that

[TABLE]

for all $t\in[0,T]$ and all $\varphi\in H^{1}_{0}(G)$, for almost all $\omega\in\Omega.$

3) If $u_{0}\in L^{2}(\Omega;H^{2}(G)\displaystyle\cap H^{1}_{0}(G)),f\in L^{2}(\Omega;L^{2}(0,T;L^{2}(G))),g\in L^{2}(\Omega;L^{2}(0,T;H^{1}(G)))$, then (2.15) has a unique solution $u\in L^{2}(\Omega;C([0,T];H^{2}(G)\displaystyle\cap H^{1}_{0}(G)))\displaystyle\cap L^{2}(\Omega,L^{2}(0,T;H^{2}(G)))$ and

[TABLE]

Moreover, it holds that

[TABLE]

for all $t\in[0,T]$ and all $\varphi\in L^{2}(G)$, for almost all $\omega\in\Omega.$

Proof.

The main idea in this part comes from [22, 15, 16, 17].

We consider the stochastic differential equation

[TABLE]

where

[TABLE]

We set

[TABLE]

it holds that

[TABLE]

as $m\rightarrow\infty.$

1. We have

[TABLE]

it follows from Itô’s rule that

[TABLE]

thus,

[TABLE]

namely, we have

[TABLE]

Taking mathematical expectation from both sides of the above inequality, we have

[TABLE]

By the Burkholder-Davis-Gundy inequality, we have

[TABLE]

thus,

[TABLE]

According to (2.29) and (2.30), we have

[TABLE]

Taking the sum on $k$ in (2.31), we get

[TABLE]

thus,

[TABLE]

where $C$ denotes a positive constant independent of $n,m$ and $T.$

Next we observe that the right-hand side of (2.33) converges to zero as $n,m\rightarrow\infty$. Hence, it follows that $\{u^{m}\}_{m=1}^{+\infty}$ is a Cauchy sequence that converges strongly in $L^{2}(\Omega,C([0,T];L^{2}(G)))$. Let $u$ be the limit, namely, we have

[TABLE]

as $m\rightarrow\infty.$

Also, it follows from (2.28) that

[TABLE]

for all $k=1,2,3\cdots$, and all $t\in[0,T]$, for almost all $\omega\in\Omega$.

By taking the limit in above equality as $m$ goes to infinity, it holds that

[TABLE]

for all $k=1,2,3\cdots$, and all $t\in[0,T]$, for almost all $\omega\in\Omega$. Thus, we have

[TABLE]

holds for all $t\in[0,T]$ and all $\varphi\in H^{2}(G)\displaystyle\cap H^{1}_{0}(G)$, for almost all $\omega\in\Omega.$

Namely, $u$ is a solution to (2.15). By taking the limit in (2.32) as $m$ goes to infinity, we can obtain (2.23).

Now, we prove the uniqueness of the solution for (2.15). Indeed, if $u_{1}$ and $u_{2}$ are the solutions for (2.15), according to (2.23), we have

[TABLE]

thus,

[TABLE]

2. Let

[TABLE]

following [23, P28] or [25], we have

[TABLE]

By multiplying (2.31) by $1+\varepsilon\lambda_{k}$, we have

[TABLE]

Taking the sum on $k$ in (2.34), we get

[TABLE]

thus,

[TABLE]

where $C$ denotes a positive constant independent of $n,m$ and $T.$ Next we observe that the right-hand side of (2.36) converges to zero as $n,m\rightarrow\infty$. Hence, it follows that $\{u^{m}\}_{m=1}^{+\infty}$ is a Cauchy sequence that converges strongly in $L^{2}(\Omega,C([0,T];H^{1}(G)))\displaystyle\cap L^{2}(\Omega,L^{2}(0,T;H^{1}(G)))$. Let $u$ be the limit, namely, we have

[TABLE]

as $m\rightarrow\infty.$

Also, it follows from (2.28) that

[TABLE]

for all $k=1,2,3\cdots$, and all $t\in[0,T]$, for almost all $\omega\in\Omega$.

By taking the limit in above equality as $m$ goes to infinity, it holds that

[TABLE]

for all $k=1,2,3\cdots$, and all $t\in[0,T]$, for almost all $\omega\in\Omega$.

Thus, it holds that

[TABLE]

holds for all $t\in[0,T]$ and all $\varphi\in H^{1}_{0}(G)$, for almost all $\omega\in\Omega.$

By taking the limit in (2.35) as $m$ goes to infinity, we can obtain (2.24).

3. We have

[TABLE]

Multiplying (2.31) by $(1+\varepsilon\lambda_{k})\lambda_{k}$, we have

[TABLE]

Taking the sum on $k$ in (2.37), we get

[TABLE]

thus,

[TABLE]

where $C$ denotes a positive constant independent of $n,m$ and $T.$ Next we observe that the right-hand side of (2.38) converges to zero as $n,m\rightarrow\infty$. Hence, it follows that $\{u^{m}\}_{m=1}^{+\infty}$ is a Cauchy sequence that converges strongly in $L^{2}(\Omega,C([0,T];H^{2}(G)))\displaystyle\cap L^{2}(\Omega,L^{2}(0,T;H^{2}(G)))$. Let $u$ be the limit.

By the same argument as in 1) and 2), $u$ is the solution of (2.15). ∎

3 Proof of Theorem 1.1

If there is no danger of confusion, we shall omit the subscript $\varepsilon,$ we use $\overline{u}_{n}$ instead of $\overline{u}_{n}^{\varepsilon}$ and $\overline{v}_{n}$ instead of $\overline{v}_{n}^{\varepsilon}.$

The proof of the existence of the weak martingale solution is divided into several steps.

Step 1. Construct the approximate solution.

Let $\{(\overline{\Omega},\overline{\mathcal{F}},\overline{\mathbb{P}}),(\overline{\mathcal{F}}_{t})_{0\leq t\leq T},\overline{B}\}$ be a fixed stochastic basis and $\{e_{n}:n=1,2,3\cdots\}$ be an orthonormal basis of $L^{2}(I)$ which was obtained in Lemma 2.5. Set $H_{n}=Span\{e_{1},e_{2},...,e_{n}\}$ and let $P_{n}$ be the $L^{2}-$orthogonal projection from $L^{2}(I)$ onto $H_{n}.$

We set

[TABLE]

and it is the solution of the following system of stochastic differential equations

[TABLE]

defined on $\{(\overline{\Omega},\overline{\mathcal{F}},\overline{\mathbb{P}}),(\overline{\mathcal{F}}_{t})_{0\leq t\leq T},\overline{B}\}.$ The mathematical expectation with respect to $\overline{\mathbb{P}}$ is denoted by $\overline{\mathbb{E}}.$

It is easy to see that $c^{n}_{k}$ satisfies the following system of stochastic differential equations

[TABLE]

By the theory of stochastic differential equations, there is a local $\overline{u}_{n}$ defined on $[0,T_{n}].$ The following a priori estimates will enable us to prove that $T_{n}=T.$

Step 2. A priori estimates.

Lemma 3.1**.**

There exists a positive constant $C$ independent of $\varepsilon$ such that

[TABLE]

for any $n\geq 1.$

Proof.

Indeed, it follows from Itô’s rule that

[TABLE]

namely, we have

[TABLE]

Taking the sum on $k$ in (3.10), following [23, P28] or [25], we get

[TABLE]

namely,

[TABLE]

It is easy to see

[TABLE]

By the Burkholder-Davis-Gundy inequality and Cauchy inequality, we can obtain that for any $\delta>0$,

[TABLE]

It follows from (3.19) that

[TABLE]

By choosing $\delta>0$ small enough, yields

[TABLE]

According to Gronwall’s lemma, we obtain that

[TABLE]

∎

The following result is related to the higher integrability of $\overline{u}_{n}.$

Lemma 3.2**.**

For any $1\leq p<\infty,$ there exists a constant $C_{p}$ independent of $\varepsilon$ such that

[TABLE]

for any $n\geq 1.$

Proof.

Case I: $2\leq p<\infty.$

To simplify the notation, we define

[TABLE]

Thus we can rewrite (3.14) as

[TABLE]

By Itô’s rule, we obtain that

[TABLE]

for any $2\leq p<\infty.$ Namely, we have

[TABLE]

Using the properties of $g$ and Young’s inequality, we have

[TABLE]

According to the Burkholder-Davis-Gundy inequality and Young’s inequality, it can be deduced that

[TABLE]

From the above estimates and (3.22), by choosing $\delta>0$ small enough, it holds that

[TABLE]

According to Gronwall’s lemma and the definition of $\phi_{n}$, we obtain that

[TABLE]

In view of (3.19), there holds

[TABLE]

Thus, we have

[TABLE]

then, for any $2\leq p<\infty,$ it holds that

[TABLE]

By the Burkholder-Davis-Gundy inequality and Young’s inequality, we have

[TABLE]

Thus

[TABLE]

According to (3.23), it holds that

[TABLE]

Case II: $1\leq p<2.$

This case can be obtained from Case I and the Young inequality. ∎

The next estimate is very important for the proof of the tightness of the law of the Galerkin solution $\{\overline{u}_{n}\}_{n\geq 1}.$

Lemma 3.3**.**

There exists a positive constant $C$ independent of $\varepsilon$ such that

[TABLE]

for any $0<\delta\leq 1.$

Remark 3.1**.**

In the above lemma, $\overline{u}_{n}$ is extended to [math] outside $[0,T]$.

Proof.

We set

[TABLE]

it is easy to see that

[TABLE]

which implies

[TABLE]

Taking the square in both side of (3.25), we have

[TABLE]

We can infer from (3.20) and (3.21) that

[TABLE]

By the Burkholder-Davis-Gundy inequality and Young’s inequality, we have

[TABLE]

It follows from (3.25)-(3.27) that

[TABLE]

By the regularity theory of elliptic equation

[TABLE]

we have

[TABLE]

thus, we have (3.24). ∎

Step 3. Tightness property of Galerkin solutions.

We may rewrite Lemma 2.1 in the following more convenient form.

By the same way as in [30, P919], according to the priori estimates (3.9)(3.20)(3.21)(3.24), we obtain that

Lemma 3.4**.**

For any $1\leq p<\infty$ and for any sequences $\mu_{m},\nu_{m}$ converging to [math] such that the series $\displaystyle\sum\limits_{m=1}^{\infty}\frac{\mu_{m}^{\frac{1}{2}}}{\nu_{m}}$ converges, $\{\overline{u}_{n}:n\in\mathbb{N}\}$ is bounded in $X_{p,\mu_{m},\nu_{m}}^{1}$ (the explicit definition of the space $X_{p,\mu_{m},\nu_{m}}^{1}$ can be found in Section 2) for any $m.$

Let

[TABLE]

and $\mathcal{B}(X)$ be the $\sigma-$algebra of the Borel sets of $X.$

For each $n,$ let $\Phi_{n}$ be the map

[TABLE]

and $\Pi_{n}$ be a probability measure on $(X,\mathcal{B}(X))$ defined by

[TABLE]

Proposition 3.1**.**

The family of probability measures $\{\Pi_{n}:n=1,2,3,...\}$ is tight in $X.$

Proof.

For any $\rho>0,$ we should find the compact subsets

[TABLE]

such that

[TABLE]

Noting the formula

[TABLE]

we define

[TABLE]

where $n\in\mathbb{N},$ $L_{\rho}$ is a constant depending on $\rho$ and will be chosen later.

By the Chebyshev inequality, we get

[TABLE]

we choose $L_{\rho}^{4}=2C\rho^{-1}\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{n^{2}}$ to get (3.28).

Let $Y_{\rho}^{1}$ be a ball of radius $M_{\rho}$ in $Y_{\mu_{m},\nu_{m}}^{1}$ (the explicit definition of the space $Y_{\mu_{m},\nu_{m}}^{1}$ can be found in Section 2), centered at zero, namely $Y_{\rho}^{1}=\{u\in Y_{\mu_{m},\nu_{m}}^{1}~{}|~{}\|u\|_{Y_{\mu_{m},\nu_{m}}^{1}}\leq M_{\rho}\}.$ From Corollary 2.1, $Y_{\rho}^{1}$ is a compact subset of $L^{2}(0,T;L^{2}(I)),$ and

[TABLE]

choosing $M_{\rho}=2C\rho^{-1},$ we get (3.29).

It follows from (3.28) and (3.29) that

[TABLE]

for any $n\geq 1.$

Thus, the family of probability measures $\{\Pi_{n}:n=1,2,3,...\}$ is tight in $X.$ ∎

Step 4. Applications of Prokhorov Theorem and Skorokhod Theorem.

By Lemma 2.3, we can find a probability measure $\Pi$ and extract a subsequence from $\Pi_{n}$ such that

[TABLE]

weakly in $X.$

By Lemma 2.4, there exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and random variables $(u_{n_{i}},B_{n_{i}}),$ $(u,B)$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in $X$ such that the probability law of $(u_{n_{i}},B_{n_{i}})$ is $\Pi_{n_{i}}.$ Furthermore,

[TABLE]

and the probability law of $(u,B)$ is $\Pi.$

Set

[TABLE]

By the idea in [30, 31], we can know $B(t)$ is a $\mathcal{F}_{t}-$standard Wiener process.

We claim that $(u_{n_{i}},B_{n_{i}})$ verifies the following $dt\otimes d\mathbb{P}-$almost everywhere:

[TABLE]

for all $\varphi\in H^{1}_{0}(I).$

Indeed, we set

[TABLE]

It is easy to see almost surely $X_{n}=0,$ hence, in particular, $\overline{\mathbb{E}}\frac{X_{n}}{1+X_{n}}=0.$

Next, we show that

[TABLE]

which will imply (3.30).

Indeed, motivated by [30], we introduce a regularization of $g,$ given by

[TABLE]

where $\beta$ is a mollifier. It is easy to check that

[TABLE]

and

[TABLE]

Then we denote by $X_{n,\rho}$ and $Y_{n_{i},\rho}$ the analog of $X_{n}$ and $Y_{n_{i}}$ with $g$ replaced by $g^{\rho}.$ Introduce the mapping

[TABLE]

owing to the definition of $X_{n,\rho},$ it is easy to see that $\Phi_{n,\rho}$ is bounded and continuous on $C([0,T],\mathbb{R}^{1})\times L^{2}(0,T;L^{2}(I)).$ Similarly, set

[TABLE]

According to Lemma 2.4, we have

[TABLE]

therefore,

[TABLE]

It is clear that

[TABLE]

As $\rho\rightarrow 0,$ it follows that

[TABLE]

It follows that (3.30) holds.

Step 5. Passage to the limit.

From (3.30), it follows that $u_{n_{i}}$ satisfies the results of (3.9)(3.20)(3.21)(3.24), we can extract from $u_{n_{i}}$ a subsequence still denoted with the same fashion and a function $u$ such that

[TABLE]

By Vitali’s convergence theorem, we have

[TABLE]

It follows from these facts that we can extract again from $u_{n_{i}}$ a subsequence still denoted by the same symbols such that

[TABLE]

It follows from (3.32) that for any $t\in[0,T],$

[TABLE]

Since $u_{n_{i}}$ is bounded in $L^{4}(\Omega,L^{4}(0,T;L^{4}(I)))$, we have $u^{3}_{n_{i}}$ is bounded in $L^{\frac{4}{3}}([0,T]\times I\times\Omega),$ Combining this and (3.33), we deduce that

[TABLE]

By (3.31), the continuity of $g,$ and the applicability of Vitali’s convergence theorem we have

[TABLE]

By the idea in [4, P284] and [30, P922], we can know

[TABLE]

for any $t\in[0,T]$.

As

[TABLE]

then

[TABLE]

Collecting all the convergence results (3.31)-(3.37), we deduce that $(u,B)$ verifies the following equation $dt\otimes d\mathbb{P}-$almost everywhere:

[TABLE]

for all $\varphi\in H^{1}_{0}(I).$

Estimates (1.2)-(1.4) follow from passing to the limits in (3.20), (3.21) and (3.24).

4 Proof of Theorem 1.2

This section is motivated by [32].

It follows from Theorem 1.1 that there exists a sequence of weak martingale solutions

[TABLE]

satisfy the inequalities

[TABLE]

where $C(p,T)$ is a constant independent of $\varepsilon.$

By the same way as in [30, P919] and [32, P2237], according to the priori estimates (4.1), we obtain that

Lemma 4.1**.**

For any $1\leq p<\infty$ and for any sequences $\mu_{m},\nu_{m}$ converging to [math] such that the series $\displaystyle\sum\limits_{m=1}^{\infty}\frac{\mu_{m}^{\frac{1}{2}}}{\nu_{m}}$ converges, $\{u^{\varepsilon}\}_{0<\varepsilon<1}$ is bounded in $X_{p,\mu_{m},\nu_{m}}^{1}$ (the explicit definition of the space $X_{p,\mu_{m},\nu_{m}}^{1}$ can be found in Section 2) for any $m.$

Let

[TABLE]

and $\mathcal{B}(X)$ be the $\sigma-$algebra of the Borel sets of $X.$

For each $\varepsilon,$ let $\Phi_{\varepsilon}$ be the map

[TABLE]

and $\Pi_{\varepsilon}$ be a probability measure on $(X,\mathcal{B}(X))$ defined by

[TABLE]

Proposition 4.1**.**

The family of probability measures $\{\Pi_{\varepsilon}:\varepsilon\in[0,1]\}$ is tight in $X.$

Proof.

We use the same method as in Proposition 3.1.

For any $\rho>0,$ we should find the compact subsets

[TABLE]

such that

[TABLE]

Noting the formula

[TABLE]

we define

[TABLE]

where $n\in\mathbb{N},$ $L_{\rho},M_{\rho}$ two constants depending on $\rho$ and will be chosen later.

By the Chebyshev inequality and the same argument as in Proposition 3.1, we get

[TABLE]

we choose $L_{\rho}^{4}=2C\rho^{-1}\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{n^{2}},M_{\rho}=2C\rho^{-1},$ to get (4.2) and (4.3).

It follows from (4.2) and (4.3) that

[TABLE]

for any $\varepsilon\in[0,1].$

Thus, the family of probability measures $\{\Pi_{\varepsilon}:\varepsilon\in[0,1]\}$ is tight in $X.$ ∎

From the tightness of $\{\Pi_{\varepsilon}:\varepsilon\in[0,1]\}$ in the Polish space $X$ and Prokhorov s theorem, we infer the existence of a subsequence $\Pi_{\varepsilon_{i}}$ of probability measures and a probability measure $\Pi$ such that $\Pi_{\varepsilon_{i}}\rightharpoonup\Pi$ weakly as $i\rightarrow\infty.$

By Lemma 2.4, there exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and random variables $(\tilde{u}^{\varepsilon_{i}},\tilde{B}^{\varepsilon_{i}}),$ $(u,B)$ on $(\Omega,\mathcal{F},\mathbb{P})$ with values in $X$ such that

[TABLE]

By the same argument as in (3.30), we have

[TABLE]

for all $\varphi\in H^{1}_{0}(I).$

From (4.4), it follows that $\tilde{u}^{\varepsilon_{i}}$ satisfies the results of (3.9)(3.20)(3.21)(3.24), we can extract from $\tilde{u}^{\varepsilon_{i}}$ a subsequence still denoted with the same fashion and a function $u$ such that

[TABLE]

By Vitali’s convergence theorem, we have

[TABLE]

according to this equality, Theorem 1.3, [4, P284], [11, P1126,Lemma 2.1] and [14, P151,Lemma 3.1], it is easy to see that for any $\delta>0,$ we have

[TABLE]

It follows from

[TABLE]

that

[TABLE]

By taking the limit in probability as $i$ goes to infinity in (4.4), we deduce that $(u,B)$ verifies the following equation $dt\otimes d\mathbb{P}-$almost everywhere:

[TABLE]

for all $\varphi\in H^{1}_{0}(I).$ Namely, $\{(\Omega,\mathcal{F},\mathbb{P}),(\mathcal{F}_{t})_{0\leq t\leq T},u,B\}$ is a weak martingale solution of problem (1.9).

5 Proof of Theorem 1.3

If there is no danger of confusion, we shall omit the subscript $\varepsilon,$ we use $u$ instead of $u^{\varepsilon}$ and $v$ instead of $v^{\varepsilon}.$

The proof is divided into several steps.

5.1 Local existence

.

Based on Proposition 2.1, we can obtain the following result.

Proposition 5.1**.**

For any $\varepsilon\in[0,\frac{1}{2}],T>0.$ If

[TABLE]

then equation

[TABLE]

has a unique solution $u^{\varepsilon}\in L^{2}(\Omega;C([0,T];H^{2}(I)\displaystyle\cap H^{1}_{0}(I)))$ and

[TABLE]

where $C=C(L,T,I).$

Proof.

The main idea in this part comes from [22].

We set

[TABLE]

$u_{n+1}(t)$ is the solution of

[TABLE]

Then,

[TABLE]

It follows from (2.26) that

[TABLE]

We define

[TABLE]

then, we have

[TABLE]

It is easy to see that

[TABLE]

which yields

[TABLE]

Consequently, $\{u_{n}\}_{n=1}^{+\infty}$ is a Cauchy sequence in $L^{2}(\Omega,C([0,T];H^{2}(I)))$. Then it is easy to see that the limit gives a solution of (5.1).

According to Proposition 2.1 (3), we have

[TABLE]

the Ironwall inequality now implies (5.2).

The uniqueness can also be obtained from the Ironwall inequality. ∎

Let $\rho\in C^{\infty}_{0}(\mathbb{R})$ be a cut-off function such that $\rho(r)=1$ for $r\in[0,1]$ and $\rho(r)=0$ for $r\geq 2.$ For any $R>0,y\in H^{1}(I)$ and $t\in[0,T],$ we set

[TABLE]

It is easy to see

[TABLE]

The truncated equation corresponding to (1.1) is the following stochastic partial differential equation:

[TABLE]

It follows from Proposition 5.1 that (5.10) has a unique solution $u_{R}\in L^{2}(\Omega;C([0,T];H^{2}(I)\displaystyle\cap H^{1}_{0}(I))).$ We define

[TABLE]

with the usual convention that $\displaystyle\inf\emptyset=+\infty.$

Since the sequence of stopping times $\tau_{R}$ is non-decreasing on $R,$ we can put

[TABLE]

We can define a local solution to (5.10) as

[TABLE]

on $[0,\tau_{R}],$ which is well defined since

[TABLE]

on $[0,\tau_{R_{1}}\wedge\tau_{R_{2}}].$

Indeed, $u_{R_{1}}(t)-u_{R_{2}}(t)$ is the solution of

[TABLE]

for $t\leq[0,\tau_{R_{1}}\wedge\tau_{R_{2}}]$ with $R_{1}\leq R_{2},$ it follows from Proposition 2.1 that

[TABLE]

where $\beta(t)$ is a continuous increasing function with $\beta(0)=0.$

If we take $t$ sufficiently small, we have $u_{R_{1}}=u_{R_{2}}$ on $[0,t].$ Repeating the same argument in the interval $[t,2t]$ and so on yields

[TABLE]

in the whole interval $[0,\tau_{R_{1}}\wedge\tau_{R_{2}}]$.

At the end, if $\tau^{*}<+\infty$, the definition of $u$ yields

[TABLE]

which shows that $u$ is a unique local solution to (5.10) on the interval $[0,\tau^{*}]$, and thus completes the proof.

5.2 Global existence

We will exploit an energy inequality.

For any $T>0,$ set $\tau=\displaystyle\inf\{\tau^{*},T\}$ and $t<\tau.$

Step 1. We first prove (1.11).

Set

[TABLE]

It follows from Itô’s rule that

[TABLE]

namely, we have

[TABLE]

After some calculation, we obtain

[TABLE]

by the Burkholder-Davis-Gundy inequality, we have

[TABLE]

By taking $\sigma<<1,\rho<<1,$ we have

[TABLE]

By the regularity theory of elliptic equation

[TABLE]

we have

[TABLE]

This implies that (1.11) holds.

Step 2. We shall prove (1.12).

According to Gagliardo-Nirenberg inequality, we have

[TABLE]

thus,

[TABLE]

In view of (1.11) and (5.11), there holds that $u^{3}-u\in L^{2}(\Omega;L^{2}(0,T;L^{2}(I))),$ moreover, $g(u)\in L^{2}(\Omega;L^{2}(0,T;H^{1}(I))),$ according to Proposition 2.1 (3), we have

[TABLE]

With the help of (1.11) and (5.11), one finds that

[TABLE]

Namely, we prove (1.12).

Step 3. We shall prove $\mathbb{P}(\{\omega\in\Omega~{}|~{}\tau^{*}(\omega)=+\infty\})=1$.

Indeed, by the Chebyshev inequality, (1.12) and the definition of $u,$ we have

[TABLE]

this show that

[TABLE]

namely, $\tau_{\infty}=+\infty$ P-a.s.

6 Proof of Theorem 1.4

6.1 A priori estimate of $\{u^{\varepsilon}\}_{0<\varepsilon<\frac{1}{2}}$

In this section, we will establish the following estimate

[TABLE]

Establishing this estimate directly for $u^{\varepsilon}$ is very difficulty, movetived by Section 2, we should establish estimate for $v^{\varepsilon},$ then by applying the regularity theory of elliptic equation, we can obtain the estimate for $u^{\varepsilon}.$

It is easy to see that

[TABLE]

which implies

[TABLE]

Taking the square in both side of (6.2), we have

[TABLE]

We can infer from (1.12) and (5.11) that

[TABLE]

By the Burkholder-Davis-Gundy inequality and Young’s inequality, we have

[TABLE]

It follows from (6.3)-(6.4) that

[TABLE]

By the regularity theory of elliptic equation

[TABLE]

we have

[TABLE]

thus, we have (6.1).

6.2 Tightness property of $\{u^{\varepsilon}\}_{0<\varepsilon<\frac{1}{2}}$ in $L^{2}(0,T;H^{1}(I))$

We may rewrite Lemma 2.1 in the following more convenient form.

By the same way as in [30, P919], according to the priori estimates (1.11)(1.12) and (6.1), we obtain that

Lemma 6.1**.**

For any $1\leq p<\infty$ and for any sequences $\mu_{m},\nu_{m}$ converging to [math] such that the series $\displaystyle\sum\limits_{m=1}^{\infty}\frac{\mu_{m}^{\frac{1}{2}}}{\nu_{m}}$ converges, $\{u^{\varepsilon}\}_{0<\varepsilon<\frac{1}{2}}$ is bounded in $X_{p,\mu_{m},\nu_{m}}^{2}$ (the explicit definition of the space $X_{p,\mu_{m},\nu_{m}}^{2}$ can be found in Section 2) for any $m.$

Set

[TABLE]

and $\mathcal{B}(S)$ the $\sigma-$algebra of the Borel sets of $S.$

For any $0<\varepsilon<\frac{1}{2},$ let $\Phi_{\varepsilon}$ be the map

[TABLE]

and $\Pi_{\varepsilon}$ be a probability measure on $(S,\mathcal{B}(S))$ defined by

[TABLE]

Proposition 6.1**.**

The family of probability measures $\{\Pi_{\varepsilon}:0<\varepsilon<\frac{1}{2}\}$ is tight in $S.$

Proof.

For any $\rho>0,$ we should find the compact subsets

[TABLE]

such that

[TABLE]

Indeed, let $Y_{\rho}^{2}$ be a ball of radius $M_{\rho}$ in $Y_{\mu_{m},\nu_{m}}^{2}$ (the explicit definition of the space $Y_{\mu_{m},\nu_{m}}^{2}$ can be found in Section 2), centered at zero and with sequences $\mu_{m},\nu_{m}$ independent of $\varepsilon,$ converging to [math] and such that the series $\displaystyle\sum\limits_{m=1}^{\infty}\frac{\mu_{m}^{\frac{1}{2}}}{\nu_{m}}$ converges. From Corollary 2.1, $Y_{\rho}^{2}$ is a compact subset of $L^{2}(0,T;H^{1}(I)),$ and

[TABLE]

choosing $M_{\rho}=C\rho^{-1},$ we get (6.5).

This proves that

[TABLE]

for any $0<\varepsilon<\frac{1}{2}.$ ∎

6.3 The convergence result

The main idea in this part comes from [6, 7].

The proof of Theorem 1.4 is divided into several steps.

Step 1. We prove that $u^{\varepsilon}$ converges in probability to some random variable $z\in L^{2}(0,T;H^{1}(I)).$

As proved in Proposition 6.1, the family $\mathcal{L}(u^{\varepsilon})$ is tight in $L^{2}(0,T;H^{1}(I))$. Then, due to the Skorokhod theorem for any two sequences $\{\varepsilon_{n}\}_{n\in N}$ and $\{\varepsilon_{m}\}_{m\in N}$ converging to zero, there exist subsequences $\{\varepsilon_{n(k)}\}_{k\in N}$ and $\{\varepsilon_{m(k)}\}_{k\in N}$ and a sequence of random elements

[TABLE]

in $L^{2}(0,T;H^{1}(I))\times L^{2}(0,T;H^{1}(I))\times C([0,T];\mathbb{R})$, defined on some probability space $(\hat{\Omega},\hat{\mathcal{F}},\hat{\mathbb{P}})$, such that

[TABLE]

namely,

[TABLE]

for each $k\in N$, and $\rho_{k}$ converges $\hat{\mathbb{P}}$-a.s. to some random element $\rho:=(u_{1},u_{2},\hat{B})\in L^{2}(0,T;H^{1}(I))\times L^{2}(0,T;H^{1}(I))\times C([0,T];\mathbb{R})$.

We now prove $u_{1}=u_{2}.$

Indeed, according to the fact that $u_{1}^{k}$ and $u_{2}^{k}$ solve (1.1) with $B$ replaced by $\hat{B}_{k},$ namely, we have

[TABLE]

and

[TABLE]

it holds that

[TABLE]

It follows from Vitali’s convergence theorem that

[TABLE]

according to this equality, Theorem 1.3, [4, P284], [11, P1126,Lemma 2.1] and [14, P151,Lemma 3.1], it is easy to see for any $\delta>0$ and any $\varphi\in H^{1}_{0}(I),$ we have

[TABLE]

By the same way, we have

[TABLE]

By taking the limit in probability as $k$ goes to infinity, we have

[TABLE]

Then, $u_{1},u_{2}$ coincide with the unique solution of heat equation perturbed by the noise $\hat{B},$ thus $u_{1}=u_{2}.$

It follows from Lemma 2.2 that $u^{\varepsilon}$ converges in probability to some random variable $z\in L^{2}(0,T;H^{1}(I))$.

Step 2. We prove that $z$ is the solution of (1.13).

It follows from

[TABLE]

that

[TABLE]

Noting that

[TABLE]

we have

[TABLE]

By taking the limit in probability as $\varepsilon$ goes to zero in

[TABLE]

we deduce that $z$ verifies the following equation $dt\otimes d\mathbb{P}-$almost everywhere:

[TABLE]

that is $z$ is the solution of (1.13).

Acknowledgements.

I sincerely thank Professor Yong Li for many useful suggestions and help.