Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains

We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.     

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier-Stokes equations

The asymptotic behavior of solutions of the three-dimensional Navier-Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray-Hopf weak solution to its weak ω-limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω-limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω-limit set are continuous in the strong topology of H. Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H.     

Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains

We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. ?ukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context.     

On the Kneser property for reaction-diffusion systems on unbounded domains

We prove the Kneser property (i.e. the connectedness and compactness of the attainability set at any time) for reaction-diffusion systems on unbounded domains in which we do not know whether the property of uniqueness of the Cauchy problem holds or not.Using this property we obtain that the global attractor of such systems is connected.Finally, these results are applied to the complex Ginzburg-Landau equation.     

Uniform attractors for the closed process and applications to the reaction-diffusion equation with dynamical boundary condition

In this paper, we first introduce the concept of a closed process in a Banach space, and we obtain the structure of a uniform attractor of the closed process by constructing a skew product-flow on the extended phase space. Then, the properties of the kernel section of closed process are investigated. Moreover, we prove the existence and structure of the uniform attractor for the reaction-diffusion equation with a dynamical boundary condition in L^p without any restriction on the growth order of the nonlinear term.     

Long-time behavior of reaction-diffusion equations with dynamical boundary condition

In this paper, we study the long-time behavior of the reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms f and g satisfy the polynomial growth condition of arbitrary order. Some asymptotic regularity of the solution has been proved. As an application of the asymptotic regularity results, we can not only obtain the existence of a global attractor A in (H¹(Ω)∩L^p(Ω))×L^q(Γ) immediately, but also can show further that A attracts every L²(Ω)×L²(Γ)-bounded subset with (H¹(Ω)∩L^p+δ(Ω))×L^q+κ(Γ)-norm for any δ,κ∈[0,∞).     

Pullback attractors for the non-autonomous FitzHugh–Nagumo system on unbounded domains

The existence of a pullback attractor is established for the singularly perturbed FitzHugh–Nagumo system defined on the entire space Rⁿ${\mathbb{R}}^n$ when external terms are unbounded in a phase space. The pullback asymptotic compactness of the system is proved by using uniform a priori estimates for far-field values of solutions. Although the limiting system has no global attractor, we show that the pullback attractors for the perturbed system with bounded external terms are uniformly bounded, and hence do not blow up as a small parameter approaches zero.     

Asymptotic behavior of the Newton–Boussinesq equation in a two-dimensional channel

We prove the existence of a global attractor for the Newton–Boussinesq equation defined in a two-dimensional channel. The asymptotic compactness of the equation is derived by the uniform estimates on the tails of solutions. We also establish the regularity of the global attractor.     

Continuity of strong solutions of the Reaction–Diffusion equation in initial data

This paper is concerned with the continuity of strong solutions of the Reaction–Diffusion equation with respect to initial conditions. It is proved that the solutions of the equation are continuous in H¹$H^1$ with respect to initial data provided the nonlinearity satisfies a dissipative condition with polynomial growth order p≥2$p\ge 2$.     

Dynamics of systems on infinite lattices

The dynamics of infinite-dimensional lattice systems is studied. A necessary and sufficient condition for asymptotic compactness of lattice dynamical systems is introduced. It is shown that a lattice system has a global attractor if and only if it has a bounded absorbing set and is asymptotically null. As an application, it is proved that the lattice reaction-diffusion equation has a global attractor in a weighted l² space, which is compact as well as contains traveling waves. The upper semicontinuity of global attractors is also obtained when the lattice reaction-diffusion equation is approached by finite-dimensional systems.     

Transversality in scalar reaction-diffusion equations on a circle

We prove that stable and unstable manifolds of hyperbolic periodic orbits for general scalar reaction-diffusion equations on a circle always intersect transversally. The argument also shows that for a periodic orbit there are no homoclinic connections. The main tool used in the proofs is Matano's zero number theory dealing with the Sturm nodal properties of the solutions.     

Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients.     

Global attractors for the wave equation with nonlinear damping

Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation utt+kg(ut)−Δu+f(u)=0 on a bounded domain Ω⊂R³ with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C₁(|s|−C₂)?|g(s)|?C₃(1+p|s|), where 1?p<5; the damping parameter is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f^′(s)|?C₄(1+q|s|), where q?2. It is remarkable that when 2<p<5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

Robust exponential attractors for singularly perturbed Hodgkin-Huxley equations

Here we consider a singular perturbation of the Hodgkin-Huxley system which is derived from the Lieberstein's model. We study the associated dynamical system on a suitable bounded phase space, when the perturbation parameter ε (i.e., the axon specific inductance) is sufficiently small. We prove the existence of bounded absorbing sets as well as of smooth attracting sets. We deduce the existence of a smooth global attractor Aε. Finally we prove the main result, that is, the existence of a family of exponential attractors {Eε} which is Hölder continuous with respect to ε.     

Pullback attractors for a semilinear heat equation on time-varying domains

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains Ot in RN are obtained from a bounded base domain O by a C²-diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.     

Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations

This paper first introduces the so-called quasi-continuous random dynamical system (RDS) on a separable Banach space. The quasi-continuity is weaker than all the usual continuities and thus is easier to check in practice. We then establish a necessary and sufficient condition for the existence of random attractors for the quasi-continuous RDS. We also give a general method to obtain the random attractors for the RDS on the Banach space Lq(D) for q?2. As an application, it is shown that the RDS generated by the stochastic reaction-diffusion equation possesses a finite-dimensional random attractor in Lq(D) for any q?2, a comparison result of fractal dimensions under the different Lq-norms is also obtained.     

Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domains

In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as $\delta\rightarrow0$ is proved.