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This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space R n . Since the standard Sobolev embeddings on R n are not compact and the spectrum of the Laplace operator on R n are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on R n is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of random attractors of stochastic wave equations with supercritical drift driven by linear multiplicative white noise defined on unbounded domains. We first establish the uniform Strichartz estimates of the solutions with respect to noise intensity, and then prove the convergence of the solutions of the stochastic equations with respect to initial data as well as noise intensity. To overcome the non-compactness of Sobolev embeddings on unbounded domains, we first utilize the uniform tail-ends estimates to truncate the solutions in a bounded domain and then employ a spectral decomposition to establish the pre-compactness of the collection of all random attractors. We finally prove the upper semicontinuity of random attractor as noise intensity approaches zero. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic behavior of complex stochastic lattice systems driven by superlinear noise. We first prove the well-posedness and the existence of weak mean random attractors of the systems. We then show the existence of periodic measures and the weak compactness of the set of all periodic measures when a parameter varies in a bounded interval. We finally examine the limiting behavior of the periodic measures and in particular prove that every limit of periodic measures of the stochastic Ginzburg–Landau lattice system must be a periodic measure of the stochastic Schrödinger lattice system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the long term behavior of the solutions of the non-autonomous Benjamin-Bona-Mahony equation driven by nonlinear colored noise with continuous coefficients defined on three-dimensional unbounded channels. The solutions of the equation are not unique and hence generate a multivalued non-autonomous random dynamical system. We first prove the measurability of the multivalued random system by the idea of weak upper semicontinuity of solutions, and then establish the existence and uniqueness of tempered pullback random attractors based on the pullback asymptotic compactness of solutions. The difficulty of the non-compactness of Sobolev embeddings on unbounded domains is overcome by the methods of the spectral decomposition inside bounded domains as well as the uniform tail-estimates of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

This paper deals with the limiting behavior of invariant measures of the stochastic delay lattice systems. Under certain conditions, we first show the existence of invariant measures of the systems and then establish the stability in distribution of the solutions. We finally prove that any limit point of a tight sequence of invariant measures of the stochastic delay lattice systems must be an invariant measure of the corresponding limiting system as the intensity of noise converges or the time-delay approaches zero. In particular, when the stochastic delay lattice systems are stable in distribution, we show the invariant measures of the perturbed systems converge to that of the limiting system. [ABSTRACT FROM AUTHOR]

This article is concerned with the well-posedness as well as long-term dynamics of a wide class of non-autonomous, non-local, fractional, stochastic FitzHugh-Nagumo systems driven by nonlinear noise defined on the entire spaceℝRN. The well-posedness is proved for the systems with polynomial drift terms of arbitrary order as well as locally Lipschitz nonlinear diffusion terms by utilizing the pathwise and mean square uniform estimates. The mean random dynamical system generated by the solution operators is proved to possess a unique weak pullback mean random attractor in a Bochner space. The existence of invariant measures is also established for the autonomous systems with globally Lipschitz continuous diffusion terms. The idea of uniform tail-estimates of the solutions in the appropriate spaces is employed to derive the tightness of a family of probability distributions of the solutions in order to overcome the non-compactness of the standard Sobolev embeddings on ℝN as well as the lack of smoothing effect on one component of the solutions. The results of this paper are new even when the fractional Laplacian is replaced by the standard Laplacian. [ABSTRACT FROM AUTHOR]

This article is concerned with the existence as well as regularity of pullback attractors for a wide class of non-autonomous, fractional, nonclassical diffusion equations with (p, q)-growth nonlinearities defined on unbounded domains. We first establish the well-posedness of solutions as well as the existence of an energy equation, and then prove the existence of a unique pullback attractor in the fractional Sobolev space H s (R N) for all s ∈ (0 , 1 ] . Finally, we show that this attractor is a bi-spatial (H s (R N) , L r (R N)) -attractor in two cases: 2 ≤ r ≤ max (p , q) for s = 1 ; and 2 ≤ r < max (p , q) for s ∈ (0 , 1). The idea of energy equations and the method of asymptotic a priori estimates are employed to establish the pullback asymptotic compactness of the solutions in H s (R N) ∩ L r (R N) in order to overcome the non-compactness of the Sobolev embedding on the unbounded domain as well as the weak dissipation of the equations. This is the first time to study the bi-spatial attractor of the equation when the initial space is H s (R N) , and the result of this article is new even in H 1 (R N) ∩ L r (R N) when the fractional Laplace operator reduces to the standard Laplace operator. [ABSTRACT FROM AUTHOR]

The skin secretions of amphibians are a rich source of bioactive peptides. We isolated chensirin-1 and chensirin-2 from the skin secretion of the Chinese frog Rana chensinensis. Sephadex-G-50 and RP-HPLC were employed to purify these peptides. The amino acid sequences of these peptides were VLPLVGNLLNDLLGE and IIPLPLGYFAKKT, respectively, as determined by Edman degradation. The molecular weights were 1578.7 and 1460.8 Da, respectively, as analyzed by HPLC-ESI-MS. The chensirin cDNA was cloned by 5′ and 3′ amplification of cDNA ends, synthesized and purified. The antibacterial activities of the chensirins were tested using minimum inhibitory concentration, the results indicated that chensirins inhibit the growth of gram-negative and gram-positive bacteria. Among them, chensirin-1 is a novel peptide with a higher antibacterial activity compared to other similar antimicrobial peptides. These low molecular weight peptides with good antimicrobial efficacy are considered potential sources for developing new antimicrobial agents to improve traditional drug resistance. [ABSTRACT FROM AUTHOR]

This paper is concerned with weak pullback mean random attractors for mean random dynamical systems defined in Bochner spaces. We first introduce the concept of weak pullback mean random attractor with respect to the weak topology of reflexive Bochner spaces and then provide a sufficient criterion for existence and uniqueness of such attractors over a complete filtered probability space. As an application, we prove the existence and uniqueness of weak pullback mean random attractors for the stochastic reaction–diffusion equations with nonlinear drift terms as well as nonlinear diffusion terms. We also establish the existence and uniqueness of such attractors for the deterministic reaction–diffusion equations with random initial data. In this case, the periodicity of the weak pullback mean random attractors is also proved whenever the external forcing terms are periodic in time. [ABSTRACT FROM AUTHOR]

In this paper we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic partial differential equations driven by a white noise. We prove that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. When the stochastic partial differential equation is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero. [ABSTRACT FROM AUTHOR]

We study asymptotic stability of a class of non-autonomous stochastic delay lattice systems driven by a multiplicative white noise. Under certain conditions, we prove such systems have a unique tempered complete quasi-solution which exponentially pullback attracts all solutions starting from a tempered random set. When the non-autonomous deterministic forcings are time-periodic, we obtain the existence, uniqueness and exponential stability of pathwise random periodic solutions for the stochastic lattice systems with delay. The convergence of the tempered complete quasi-solution (periodic solution) is also established when time delay approaches zero. [ABSTRACT FROM AUTHOR]

We study global asymptotic behavior of Poisson–Nernst–Planck (PNP) systems for flow of two ion species through a narrow tubular-like membrane channel. As the radius of the cross-section of the three-dimensional tubular-like membrane channel approaches zero, a one-dimensional limiting PNP system is derived. This one-dimensional limiting system differs from previously studied one-dimensional PNP systems in that it encodes the defining geometry of the three-dimensional membrane channel. To justify this limiting process, we show that the global attractors of the three-dimensional PNP systems are upper semi-continuous as the radius of the channel tends to zero. [ABSTRACT FROM AUTHOR]

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L 2(ℝ n ) and H 1(ℝ n ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space ℝ n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L 2(ℝ n ) and H 1(ℝ n ), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains. [ABSTRACT FROM AUTHOR]

In the present paper, we deal with the long-time behavior of dissipative partial differential equations, and we construct the approximate inertial manifolds for the nonlinear Schrödinger equation with a zero order dissipation. The order of approximation of these manifolds to the global attractor is derived. [ABSTRACT FROM AUTHOR]

Examines the theory for the two-dimensional derivative Ginzburg-Landau equation. Nonlinear physical phenomenon that produced generalized Ginzburg-Landau equation; Approximate solution sequence by Galerkin method; Obtaining a priori estimates.

Abstract In this paper, we establish the global fast dynamics for the derivative Ginzburg-Landau equation in two spatial dimensions. We show the squeezing property and the existence of finite dimensional exponential attractors for this equation [ABSTRACT FROM AUTHOR]