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We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.

In this article we study the Borel probability measures that can be associated to the time averaged observation of the process generated by the non-autonomous long-wave-short-wave resonance equations on infinite lattices, via the notion of generalized Banach limit. We establish that the generated process possesses a pullback- D attractor, and further prove that there exists a unique family of invariant Borel probability measures carried by the pullback attractor.

This paper studies the trajectory asymptotic behavior of a non-autonomous incompressible non-Newtonian fluid in 3D bounded domains. In appropriate topologies, the authors prove the existence of the uniform trajectory attractor for the translation semigroup acting on the united trajectory space.

This paper studies the time decay rate of weak solutions to the following two-dimensional magnetohydrodynamics (MHD) equations with fractional dissipations ∂ t u + ( u · ∇ ) u − ( b · ∇ ) b + ∇ p = − ( − ▵ ) α u , ∂ t b + ( u · ∇ ) b − ( b · ∇ ) u = − ( − ▵ ) β b . The motivation is to understand how the parameters α and β affect the decay rate of its solutions. The authors use the Fourier splitting method of Schonbek to prove that the solutions have the following decay rate ∥ u ( x , t ) ∥ 2 + ∥ b ( x , t ) ∥ 2 ⩽ c ( 1 + t ) 1 − 2 / γ , for large enough t , where α, β ∈ [1, 2) and γ = max { α , β } .

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the hyperbolic magnetohydrodynamics (MHD) equations in ℝ 3 . Then we establish that the solutions with initial data belonging to H m ( ℝ 3 ) ∩ L 1 ( ℝ 3 ) have the following time decay rate: ‖ ∇ m u ( x , t ) ‖ 2 + ‖ ∇ m b ( x , t ) ‖ 2 + ‖ ∇ m + 1 u ( x , t ) ‖ 2 + ‖ ∇ m + 1 b ( x , t ) ‖ 2 ≤ c ( 1 + t ) - 3 / 2 - m for large t , where m = 0, 1.

In this article, we construct the trajectory statistical solution for the 3D incompressible Navier–Stokes equations via the natural translation semigroup and trajectory attractor. In our construction, the trajectory statistical solution is an invariant space–time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space, and the trajectory statistical solution possesses the invariant property under the acting of the translation semigroup.

In this paper, we first show the global existence, uniqueness and regularity of weak solutions for the Navier-Stokes-Voigt equations in R 3 . Then we combine the Fourier splitting method of Schonbek and the Gronwall inequality to prove that the solutions have the following decay rates ? ? m u ( x , t ) ? 2 + ? ? m + 1 u ( x , t ) ? 2 ≤ c ( 1 + t ) - 3 / 2 - m , for large t when u 0 ? H m ( R 3 ) ? L 1 ( R 3 ) and m = 0 , 1 .

In this paper, we investigate the complex dynamics of a Leslie–Gower predation model with additive Allee effect on prey. Without Allee effect, the model has a unique globally asymptotically stable equilibrium point under some conditions. With Allee effect on prey, the model has none, one or two positive equilibria. Moreover, we prove the existence of parameter subsets for which the model can have Hopf bifurcation. And we find that when the model exhibits two positive equilibria there is a separatrix curve that separates the behavior of trajectories of the system, implying that the model is highly sensitive to the initial conditions. One of the most interesting findings is that Allee effect can increase the risk of ecological extinction.

This paper discusses the large time behaviors of solutions for a class of incompressible third grade fluid equations in R 3 . Using the Fourier splitting method of Schonbek, the authors prove the upper and lower bounds of the time decay rate in L 2 for the weak solutions. The upper and lower bounds of decay rate are optimal in the sense that they coincide with the upper and lower bounds of the decay rate of solutions to the heat equation.

This paper studies the asymptotic behavior of solutions for the discrete coupled nonlinear Schrodinger-Boussinesq equations. The authors first prove the existence of a global attractor for the generated semigroup and then obtain an upper bound of the Kolmogorov @e-entropy for the obtained global attractor. Finally, they establish the upper semicontinuity of the global attractor when the infinite lattice systems are approximated by finite lattice systems.

Consider the following first order lattice system u m + ( 2 u m − u m − 1 − u m + 1 ) + λ m u m + f m ( u m ) = g m , m ∈ Z , which is perturbed by the ϵ -small two order term ϵ u m + u m + ( 2 u m − u m − 1 − u m + 1 ) + λ m u m + f m ( u m ) = g m , m ∈ Z . Under certain conditions on f m , λ m and g m , the original systems and the ϵ -small perturbed systems have global attractors A in l 2 and A ϵ in l 2 × l 2 , respectively, and A can be naturally embedded into a compact set A 0 in l 2 × l 2 . We prove the upper semicontinuity of A 0 with respect to the attractors A ϵ at zero by showing that for any neighborhood O ( A 0 ) of A 0 , A ϵ enters O ( A 0 ) if ϵ is small enough.

This paper studies the dynamical behavior of the Ladyzhenskaya model with additive noise. With some conditions, we prove that the generated random dynamical system has a compact random attractor, which is a random compact set absorbing any bounded nonrandom subset of the phase space.

There are two results within this paper. The one is the regularity of trajectory attractor and the trajectory asymptotic smoothing effect of the incompressible non-Newtonian fluid on 2D bounded domains, for which the solution to each initial value could be non-unique. The other is the upper semicontinuity of global attractors of the addressed fluid when the spatial domains vary from Ω m to Ω = R × ( − L , L ) , where { Ω m } m = 1 ∞ is an expanding sequence of simply connected, bounded and smooth subdomains of Ω such that Ω m → Ω as m → + ∞ . That is, let A and A m be the global attractors of the fluid corresponding to Ω and Ω m , respectively, we establish that for any neighborhood O ( A ) of A , the global attractor A m enters O ( A ) if m is large enough.

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.

In this paper, we consider the existence, upper semi-continuity and singular limiting behavior of compact kernel sections for the process associated to non-autonomous lattice systems with delays.

This paper discusses the incompressible non-Newtonian fluid with rapidly oscillating external forces gɛ(x,t)=g(x,t,t/ɛ) possessing the average g0(x,t) as ɛ→0+, where 0

We discuss the long time behavior of solutions for a two-dimensional (2D) nonautonomous incompressible non-Newtonian fluid in 2D bounded domains. When the external force g 0 ( x , t ) is translation compact (tr.c.) in L loc 2 ( R ; H ) , we obtain the existence and reveal the structure of the uniform attractor for the processes associated to the fluid by constructing skew product flow on the extended phase space. When ‖ g 0 ‖ L b 2 is properly small, we establish that there exists a unique bounded asymptotically stable solution to the fluid.

In this work, the authors first show the existence of global attractors A e for the following lattice complex Ginzburg–Landau equation: i u m − ( α − i e ) ( 2 u m − u m + 1 − u m − 1 ) + i κ u m + β | u m | 2 σ u m = g m , m ∈ Z , e > 0 , and A 0 for the following lattice Schrodinger equation: i u m − α ( 2 u m − u m + 1 − u m − 1 ) + i κ u m + β | u m | 2 σ u m = g m , m ∈ Z . Then they prove that the solutions of the lattice complex Ginzburg–Landau equation converge to that of the lattice Schrodinger equation as e → 0 + . Also they prove the upper semicontinuity of A e as e → 0 + in the sense that lim e → 0 + dist l 2 ( A e , A 0 ) = 0 .

This paper discusses the long time behavior of solutions for a two-dimensional (2D) nonautonomous micropolar fluid flow in 2D unbounded domains in which the Poincare inequality holds. We use the energy method to obtain the so-called asymptotic compactness of the family of processes associated with the fluid flow and establish the existence of H1-uniform attractor. Then we prove that an L2-uniform attractor belongs to the H1-uniform attractor, which implies the asymptotic smoothing effect for the fluid flow in the sense that the solutions become eventually more regular than the initial data.

In the present paper, we consider the nonautonomous long-wave–short-wave resonance equations on infinite lattices. We first prove the existence of compact kernel sections for the associated process. Then we give an upper bound of the Kolmogorov e -entropy and verify the upper semicontinuity of these kernel sections.

We prove the upper semicontinuity of the global attractor corresponding to a class of lattice nonclassical parabolic equations.

This paper discusses the Cauchy problem of wave equation with linear damping and source terms. The author reduces the problem to a parabolic one and uses the corresponding initial value to construct the solutions of the resulting parabolic problem. Thereafter, he obtains a brief algorithmization formula to this problem with mechanization in Maple.

The present paper discusses the algorithm and mechanization to the Cauchy problem of parabolic equation. The author uses the initial data to construct the solutions that satisfy the initial condition and obtain a brief algorithm to this problem. Thereafter, the algorithm is used in Maple to solve this Cauchy problem automatically.

This paper discusses the constructive method and mechanization to the Cauchy problem of high dimensional wave equations. The author uses the initial value to construct the solutions that satisfy the initial conditions and obtain a brief algorithmization formula to this problem. Thereafter, the algorithmization formula is used in Maple to solve this Cauchy problem automatically.

In this note, the authors consider the regularity of the global attractor for a non-Newtonian system in two-dimensional unbounded domains. They prove that the L 2 -compact attractor and the H 2 -compact attractor of the system are the same and thus establish the asymptotic smoothing effect of solutions to this system.

In this paper, the authors study the long time behavior of a non-Newtonian system in two-dimensional unbounded domains. They prove the existence of H2-compact attractor for the system by showing the corresponding semigroup is asymptotically compact.

This paper studies the asymptotic behavior of solutions for the three-component reversible Gray–Scott model on infinite lattices. We first prove the existence of a global attractor for the generated semigroup. Then we give an upper bound of the Kolmogorov e -entropy for the obtained global attractor.