Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Uniform exponential attractors for second order non-autonomous lattice dynamical systems

In this paper, the existence of a uniform exponential attractor for a second order non-autonomous lattice dynamical system with quasiperiodic symbols acting on a closed bounded set is considered. Firstly, the existence and uniqueness of solutions for the considered systems which generate a family of continuous processes is shown, and the existence of a uniform bounded absorbing sets for the processes is proved. Secondly, a semigroup defined on a extended space is introduced, and the Lipschitz continuity, α-contraction and squeezing property of this semigroup are proved. Finally, the existence of a uniform exponential attractor for the family of processes associated with the studied lattice dynamical systems is obtained.     

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.     

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.     

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.     

Long-Time Behavior for Second Order Lattice Dynamical Systems

Many researchers examined the existence of global attractors for various types of first and second order lattice dynamical systems. Here we prove the existence of a global attractor for a new type of second order lattice dynamical systems in the Hilbert space l ²×l ². For specific choices of the linear operators this system can be regraded as a spatial discretization of a continuous damped nonlinear Boussinesq equation on ℝ ^m ,m≥1.      

Cauchy-type determinants and integrable systems

It is well known that the Sylvester matrix equation AX + XB = C has a unique solution X if and only if 0 ∉ spec(A) + spec(B). The main result of the present article are explicit formulas for the determinant of X in the case that C is one-dimensional. For diagonal matrices A, B, we reobtain a classical result by Cauchy as a special case.The formulas we obtain are a cornerstone in the asymptotic classification of multiple pole solutions to integrable systems like the sine-Gordon equation and the Toda lattice. We will provide a concise introduction to the background from soliton theory, an operator theoretic approach originating from work of Marchenko and Carl, and discuss examples for the application of the main results.     

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,∞).     

Exponential Attractors for Lattice Dynamical Systems in Weighted Spaces

The aim of this paper is to investigate the existence of exponential attractors for lattice reaction-diffusion systems in weighted spaces l_s²l_{\sigma}^{2} and for partly dissipative lattice reaction-diffusion systems in weighted spaces l_m²×l_m²l_{\mu}^{2}\times l_{\mu}^{2}, respectively. In contrast to the previous work by Abdallah in J. Math. Anal. Appl. 339, 217–224 (2008) and Commun. Pure Appl. Anal. 8, 803–818 (2009), we get the existence of exponential attractors for lattice dynamical systems in the weak topology spaces.     

Attractors and approximations for lattice dynamical systems

We present a sufficient condition for the existence of a global attractor for general lattice dynamical systems, then consider the existence of attractors and their approximation for second-order and first-order lattice systems which, in particular case, can be regarded as the spatial discretizations of corresponding wave equations and reaction-diffusion equations in R^k.     

Convergence rate of the solution toward boundary layer solution for initial-boundary value problem of the 2-D viscous conservation laws

In this paper, we study the large time behavior of the solution to the initial boundary value problem for 2-D viscous conservation laws in the space x ? bt. The global existence and the asymptotic stability of a stationary solution are proved by Kawashima et al. [1]. Here, we investigate the convergence rate of solution toward the boundary layer solution with the non-degenerate case where f′(u₊) − b < 0. Based on the estimate in the H² Sobolev space and via the weighted energy method, we draw the conclusion that the solution converges to the corresponding boundary layer solution with algebraic or exponential rate in time, under the assumption that the initial perturbation decays with algebraic or exponential in the spatial direction.     

Stabilities in multi-valued dynamical systems

For the study of the various stabilities in multi-valued dynamical systems, we obtain some necessary and sufficient conditions of the notions of stability and Lyapunov stability, and investigate the connection between the concepts of attractions and the suitable versions of limit sets. Also we consider the notion of characteristic 0⁺ in multi-valued dynamical systems and obtain necessary and sufficient conditions for the concept of characteristic 0⁺.     

Center manifolds for difference equations—smooth parameter dependence

For sufficiently small C¹ perturbations of (nonautonomous) linear difference equations with a nonuniform exponential trichotomy, we establish the existence of center manifolds with the optimal C¹ regularity. We also consider the case of parameter-dependent perturbations and we obtain the C¹ dependence of the center manifolds on the parameter. In addition, we consider arbitrary growth rates with the usual exponential estimates of the form in the notion of exponential trichotomy replaced by where ρ is now an arbitrary function. The proof of the regularity, both of the center manifold and of its dependence, on the parameter is based on the fiber contraction principle. The most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument.     

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.     

格点系统存在指数吸引子的充分条件及应用

本文给出了一般格点动力系统存在指数吸引子的充分条件,然后将得到的结果应用到下面的格点非线性Schr(o|¨)dinger方程:iu_m-γ(2u_m-u_(m+1)-u_(m-1))+iκu_m+δ|u_m|~(2σ)u_m=g_m,m∈Z.设γ,κ,δ,σ和g_m满足适当的条件,证明了该格点方程存在指数吸引子.     

Robust exponential attractors for a phase-field system with memory

H.G. Rotstein et al. proposed a nonconserved phase-field system characterized by the presence of memory terms both in the heat conduction and in the order parameter dynamics. These hereditary effects are represented by time convolution integrals whose relaxation kernels k and h are nonnegative, smooth and decreasing. Rescaling k and h properly, we obtain a system of coupled partial integrodifferential equations depending on two relaxation times ɛ and σ. When ɛ and σ tend to 0, the formal limiting system is the well-known nonconserved phase-field model proposed by G. Caginalp. Assuming the exponential decay of the relaxation kernels, the rescaled system, endowed with homogeneous Neumann boundary conditions, generates a dissipative strongly continuous semigroup S_{ɛ, σ}(t) on a suitable phase space, which accounts for the past histories of the temperature as well as of the order parameter. Our main result consists in proving the existence of a family of exponential attractors for S_{ɛ, σ}(t), with ɛ, σ ∈ [0, 1], whose symmetric Hausdorff distance from tends to 0 in an explicitly controlled way.     

Smooth center manifolds for nonuniformly partially hyperbolic trajectories

We establish the existence of unique smooth center manifolds for ordinary differential equations v^′=A(t)v+f(t,v) in Banach spaces, assuming that v^′=A(t)v admits a nonuniform exponential trichotomy. This allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories. In addition, we prove that the center manifolds are as regular as the vector field. Our proof of the Ck smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space. To the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting.     

Stable manifolds for nonautonomous equations without exponential dichotomy

We consider nonautonomous ordinary differential equations v^′=A(t)v in Banach spaces and, under fairly general assumptions, we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v^′=A(t)v+f(t,v), which corresponds to the set of negative Lyapunov exponents of the original linear equation. The main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity, i.e., a small deviation from the classical notion of (uniform) exponential dichotomy. In fact, we showed that essentially any linear equation v^′=A(t)v admits a nonuniform exponential dichotomy and thus, the above assumption only concerns the smallness of the nonuniformity of the dichotomy. This smallness is a rather common phenomenon at least from the point of view of ergodic theory: almost all linear variational equations obtained from a measure-preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity. We emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small, only sufficiently small. Our approach is related to the notion of Lyapunov regularity, which goes back to Lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations.     

Lyapunov functions for trichotomies with growth rates

We consider linear equations x^′=A(t)x that may exhibit stable, unstable and central behaviors in different directions, with respect to arbitrary asymptotic rates ecρ(t) determined by a function ρ(t). For example, the usual exponential behavior with ρ(t)=t is included as a very special case, and when ρ(t)=logt we obtain a polynomial behavior. We emphasize that we also consider the general case of nonuniform exponential behavior, which corresponds to the existence of what we call a ρ-nonuniform exponential trichotomy. This is known to occur in a large class of nonautonomous linear equations. Our main objective is to give a complete characterization in terms of strict Lyapunov functions of the linear equations admitting a ρ-nonuniform exponential trichotomy. This includes criteria for the existence of a ρ-nonuniform exponential trichotomy, as well as inverse theorems providing explicit strict Lyapunov functions for each given exponential trichotomy. In the particular case of quadratic Lyapunov functions we show that the existence of strict Lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the Lyapunov functions. As an application of the characterization of nonuniform exponential trichotomies in terms of strict Lyapunov functions, we establish the robustness of ρ-nonuniform exponential trichotomies under sufficiently small linear perturbations. We emphasize that in comparison with former works, our proof of the robustness is much simpler even when ρ(t)=t.     

Gibbs–Markov–Young structures with (stretched) exponential tail for partially hyperbolic attractors

In this work we extend the results obtained by Gouëzel in [12] to partially hyperbolic attractors. We study a forward invariant set K on a Riemannian manifold M   whose tangent space splits as dominated decomposition _TKM=E^cu⊕E^s$T_{K}M=E^{cu}\oplus E^s$, for which the center-unstable direction E^cu$E^{cu}$ is non-uniformly expanding on some local unstable disk. We prove that the (stretched) exponential decay of recurrence times for an induced scheme can be deduced under the assumption of (stretched) exponential decay of the time that typical points need to achieve some uniform expanding in the center-unstable direction. As an application of our results we obtain exponential decay of correlations and exponential large deviations for a class of partially hyperbolic diffeomorphisms considered in [1].