Uniform attractor for 2D magneto-micropolar fluid flow in some unbounded domains

We investigate the flow of a magneto-micropolar fluid in an arbitrary unbounded domain on which the Poincaré inequality holds. Assuming homogeneous boundary conditions and the external fields to be almost periodic in time we prove the existence of the uniform attractor by using the energy method [10] which we generalize to nonautonomous systems. We consider the problem in an abstract setting that allows to include also other hydrodynamical models. In particular, we extend the result of R. Rosa [12] from autonomous to nonautonomous Navier-Stokes equations in unbounded domains.     

Bounded or unbounded solutions of a functional equation with nonautonomous iteration

Aequationes mathematicae - Motivated by nonautonomous difference equations, we study a functional equation with nonautonomous iteration of order n for bounded solutions and unbounded solutions. We...     

STABILITY THEOREM FOR A CLASS OF PERTURBED NONAUTONOMOUS NEUTRAL DIFFERENTIAL EQUATION WITH UNBOUNDED DELAY

In this paper, we consider a one-dimensional nonautonomous neutral differential equation. We obtain sufficient conditions under which the zero solution to this equation with unbounded delay and perturbation is uniformly asymptotically stable.     

Pointwise gradient estimates for evolution operators associated with Kolmogorov operators

We determine sufficient conditions for the occurrence of a pointwise gradient estimate for the evolution operators associated with nonautonomous second order parabolic operators with (possibly) unbounded coefficients. Moreover, we exhibit a class of operators which satisfy our conditions.     

Permanence of nonautonomous Lotka–Volterra delay differential systems

In this work, applying the results offered by S. Ahmad and A.C. Lazer [On a property of nonautonomous Lotka–Volterra competition model, Nonlinear Anal. 37 (1999) 603–611] and the recent work of R. Redheffer [Mean values and the nonautonomous May–Leonald equations, Nonlinear Anal. Real World Appl. 4 (2003) 301–306] to an nonautonomous Lotka–Volterra differential system with finite delays, we establish sufficient conditions for the permanence of the system.     

Attractors for 2D-Navier–Stokes equations with delays on some unbounded domains

We prove the existence of tempered and nontempered pullback attractors for two dimensional Navier–Stokes equations on unbounded domains satisfying Poincaré inequality, for the case in which a forcing term involving memory effects appears. Our proof uses an energy method and is valid for the autonomous and nonautonomous cases.     

Unbounded critical points for a class of lower semicontinuous functionals

For a general class of lower semicontinuous functionals, we prove existence and multiplicity of critical points, which turn out to be unbounded solutions to the associated Euler equation. We apply a nonsmooth critical point theory developed in [10], [12] and [13] and applied in [8], [9] and [20] to treat the case of continuous functionals.     

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.     

Cauchy problem for nonautonomous Kolmogorov equations

We study the existence and uniqueness of the strict solutions of an initial value problem for a nonautonomous Kolmogorov equation in a space of unbounded functions     

Hausdorff dimension estimation for attractors of nonautonomous dynamical systems in unbounded domains: An example

The subject of this paper is the asymptotic behavior of a class of nonautonomous, infinite‐dimensional dynamical systems with an underlying unbounded domain. We present an approach that is able to overcome both the law of compactness of the trajectories and the continuity of the spectrum of the linear part of the equations under consideration, providing nevertheless existence of uniform attractors. Moreover, our approach allows us to estimate the Hausdorff dimension of attractors of nonautonomous equations in terms of physical parameters. © 2000 John Wiley & Sons, Inc.     

Evolution Systems for Differential Equations with Variable Time Lags

We consider nonautonomous retarded functional differential equations under hypotheses which are designed for the application to equations with variable time lags, which may be unbounded, and construct an evolution system of solution operators which are continuously differentiable. These operators are defined on manifolds of continuously differentiable functions. The results apply to pantograph equations and to nonlinear Volterra integro-differential equations, for example. For linear equations we also provide a simpler evolution system with solution operators on a Banach space of continuous functions.     

Parabolic equations with nonlinear time-dependent boundary conditions

Summary We consider a parabolic equation with a nonlinear time-dependent boundary condition, where the nonlinearity is subjected only to «one-sided» conditions. In order to solve this equation, we extend some results of [6] and [9] to the nonautonomous case.Lavoro eseguito nell'ambito di un progetto nazionale di ricerca finanziato dal Ministero della Pubblica Istruzione (40% — 1983).     

The stability of stochastic partial differential equations II

We prove general results on stability (in finite time intervals) of SPDEs (stochastic partial differential equations) with unbounded coefficients, with respect to the simultaneous perturbations of the driving semimartingales, of all data, and of the underlying probability space. Hence we derive support theorems for SPDEs (with unbounded coefficients). In particular, we get theorems on supports and theorems on robustness for the nonlinear filter of diffusion processes with unbounded drift and diffusion coefficients. (The above results were proved in the case of bounded coefficients in our earlier papers [4] and [5].) Finally we treat an application in a problem of kinematic dynamo     

On Singular Perturbations of Quantum Dynamical Semigroups

We consider two examples of quantum dynamical semigroups obtained by singular perturbations of a standard generator which are special case of unbounded completely positive perturbations studied in detail in [11]. In Sec. 2, we propose a generalization of an example in [15] aimed to give a positive answer to a conjecture of Arveson. In Sec. 3 we consider in greater detail an improved and simplified construction of a nonstandard dynamical semigroup outlined in our short communication [12].     

Global bifurcation of homoclinic trajectories of discrete dynamical systems

We prove the existence of an unbounded connected branch of nontrivial homoclinic trajectories of a family of discrete nonautonomous asymptotically hyperbolic systems parametrized by a circle under assumptions involving topological properties of the asymptotic stable bundles.     

Permanence for nonautonomous Lotka–Volterra cooperative systems with delays

In this paper, we consider a class of nonautonomous two species Lotka–Volterra cooperative population systems with time delays, and establish sufficient conditions which ensure the system to be permanent. We improve and extend the known condition of the permanence in [G. Lu and Z. Lu, Permanence for two species Lotka–Volterra cooperative systems with delays, Math. Biosci. Eng. 5 (2008) 477–484] to nonautonomous two-species Lotka–Volterra cooperative systems. Moreover, our conditions need no restriction on the size of time delays.     

Random Attractor for Non-autonomous Stochastic Strongly Damped Wave Equation on Unbounded Domains

In this paper we study the asymptotic dynamics for the nonautonomous stochastic strongly damped wave equation driven by additive noise defined on unbounded domains. First we introduce a continuous cocycle for the equation and then investigate the existence and uniqueness of tempered random attractors which pullback attract all tempered random sets.     

A class of impulsive nonautonomous differential equations and Ulam–Hyers–Rassias stability

In this paper, we study a model described by a class of impulsive nonautonomous differential equations. This new impulsive model is more suitable to show dynamics of evolution processes in pharmacotherapy than the classical one. We apply Krasnoselskii's fixed point theorem to obtain existence of solutions. Meanwhile, we mainly present the sufficient conditions on Ulam–Hyers–Rassias stability on both compact and unbounded intervals. Many analysis techniques are used to derive our results. Copyright © 2014 John Wiley & Sons, Ltd.     

Infinite Element Approximation to Axial Symmetric Stokes Flow

We considered in [1] the finite element approximation to axial symmetric Stokes flow in a bounded domain. The problem for the flow passing an obstacle in an unbounded domain is also frequently encountered. In this paper, we are going to give approximate solutions for this problem by an approach stated in [2]. An iterative method is used to calculate the combined stiffness matrix.     

Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order

We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1363–1375, October, 2007.