DCDS
Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions
Boling Guo Bixiang Wang
In the present paper , we show the Gevrey class regularity of solutions for the generalized Ginzburg-Landau equation in two spatial dimensions. We also introduce an approximate inertial manifold for this system.
keywords: global attractor approximate inertial manifold Ginzburg-Landau equation. Gevrey class regularity
DCDS
Random attractors for non-autonomous stochastic wave equations with multiplicative noise
Bixiang Wang
This paper is concerned with the asymptotic behavior of solutions of the damped non-autonomous stochastic wave equations driven by multiplicative white noise. We prove the existence of pullback random attractors in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ when the intensity of noise is sufficiently small. We demonstrate that these random attractors are periodic in time if so are the deterministic non-autonomous external terms. We also establish the upper semicontinuity of random attractors when the intensity of noise approaches zero. In addition, we prove the measurability of random attractors even if the underlying probability space is not complete.
keywords: periodic attractor Random attractor stochastic wave equation. upper semicontinuity random complete solution
DCDS-B
Multivalued non-autonomous random dynamical systems for wave equations without uniqueness
Bixiang Wang

This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $\mathbb{R}^n$ is overcome by the uniform estimates on the tails of solutions.

keywords: Random attractor multivalued system measurability wave equation unbounded domain
DCDS
Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems
Bixiang Wang
In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.
keywords: stochastic bifurcation. random periodic solution Pullback attractor random automorphic solution random almost periodic solution
DCDS
Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity
Bixiang Wang Shouhong Wang
We prove in this article the Gevrey class regularity and time-analyticity of the global (in time) strong solutions obtained by Tang and Wang (1995) for the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity with an applied field.
keywords: Gevrey class regularity superconductivity global strong solutions Time-dependent Ginzburg-Landau equations time-analyticity.
DCDS-B
Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing
Abiti Adili Bixiang Wang
This paper is concerned with the asymptotic behavior of solutions of the FitzHugh-Nagumo system on $\mathbb{R}^n$ driven by additive noise and deterministic non-autonomous forcing. We prove the system has a random attractor which pullback attracts all tempered random sets. We also prove the periodicity of the random attractor when the system is perturbed by time periodic forcing. The pullback asymptotic compactness of solutions is established by uniform estimates on the tails of solutions outside a large ball in $\mathbb{R}^n$.
keywords: periodic attractor random complete solution FitzHugh-Nagumo system Pullback attractor unbounded domain.
PROC
Random attractors for non-autonomous stochastic FitzHugh-Nagumo systems with multiplicative noise
Abiti Adili Bixiang Wang
In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system defined on $\mathbb{R}^n$ driven by both deterministic non-autonomous forcing and multiplicative noise. The periodicity of random attractors is established when the system is perturbed by time periodic forcing. We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero.
keywords: upper semicontinuity unbounded domain FitzHugh-Nagumo system. Random attractor periodic attractor
PROC
Random attractors for wave equations on unbounded domains
Bixiang Wang Xiaoling Gao
The asymptotic behavior of stochastic wave equations on $\mathbb{R}^n$ is studied. The existence of a random attractor for the corresponding random dynamical system in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ is established, where the nonlinearity has an arbitrary growth order for $n \le 2$ and is subcritical for $n=3$.
keywords: Random attractor asymptotic compactness wave equation
DCDS
Limiting behavior of dynamics for stochastic reaction-diffusion equations with additive noise on thin domains
Dingshi Li Kening Lu Bixiang Wang Xiaohu Wang

In this paper, we study the limiting behavior of dynamics for stochastic reaction-diffusion equations driven by an additive noise and a deterministic non-autonomous forcing on an (n+1)-dimensional thin region when it collapses into an n-dimensional region. We first established the existence of attractors and their properties for these equations on (n+1)-dimensional thin domains. We then show that these attractors converge to the random attractor of the limit equation under the usual semi-distance as the thinness goes to zero.

keywords: Thin domain stochastic reaction-diffusion equation pullback attractor upper semicontinuity
DCDS-B
Asymptotic behavior of random Fitzhugh-Nagumo systems driven by colored noise
Anhui Gu Bixiang Wang

In this paper, we prove the existence and uniqueness of random attractors for the FitzHugh-Nagumo system driven by colored noise with a nonlinear diffusion term. We demonstrate that the colored noise is much easier to deal with than the white noise for studying the pathwise dynamics of stochastic systems. In addition, we show the attractors of the random FitzHugh-Nagumo system driven by a linear multiplicative colored noise converge to that of the corresponding stochastic system driven by a linear multiplicative white noise.

keywords: Random attractor colored noise white noise FitzHugh-Nagumo system

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