Nonautonomous Equations Research Articles

Многомерное неавтономное эволюционное уравнение типа Монжа - Ампера

Исследовано многомерное неавтономное эволюционное уравнение типа Монжа~--- Ампера. Левая часть уравнения содержит первую производную по времени с коэффициентом, зависящим от времени, пространственных переменных и искомой функции, а правая часть~--- определитель матрицы Гессе. Получены решения данного уравнения с аддитивным и мультипликативным разделением переменных, и показано, что достаточным условием существования таких решений является возможность представления коэффициента при производной по времени в виде произведения функций от времени и от пространственных переменных. Также найдены решения в виде квадратичных полиномов по пространственным координатам в случае, когда коэффициент при производной по времени имеет вид функции, обратной линейной комбинации пространственных переменных с коэффициентами, зависящими от времени. Получено множество решений в виде разложения по функциям, зависящим от подмножеств пространственных переменных с коэффициентами, зависящими от времени, и найдены достаточные условия существования таких решений. Рассмотрены некоторые редукции исходного уравнения к обыкновенным дифференциальным уравнениям (ОДУ) в случаях, когда искомая функция зависит от суммы функций пространственных координат (в частности, суммы их квадратов) и функции времени; при этом используется функциональное разделение переменных. Также найдены редукции исходного уравнения к уравнениям в частных производных меньшей размерности. В частности, получены решения в виде функции времени и суммы квадратов пространственных координат, а также в виде суммы нескольких таких функций и найдены достаточные условия их существования.

Numerical Dynamics of Integrodifference Equations: Hierarchies of Invariant Bundles in Lp (Ω)

We study how the “full hierarchy” of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of p-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of -functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the -topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fashion and technically results from a general perturbation theorem on the existence of invariant bundles (i.e. nonautonomous invariant manifolds).

Input‐to‐state practical stability for nonautonomous nonlinear infinite‐dimensional systems

AbstractIn this article, we are concerned with the input‐to‐state practical stability (ISpS) of nonautonomous nonlinear infinite‐dimensional systems. Sufficient conditions of ISpS are provided based on Lyapunov functions and a nonlinear inequality. We characterize ISpS in terms of uniform practical asymptotic gain property. We show that for a class of admissible inputs the existence of an ISpS‐Lyapunov function implies the ISpS of a system in Banach spaces. These results are used to study the input‐to‐state practical stability of a certain class of nonautonomous semilinear evolution equations. Furthermore, we provide a small‐gain theorem for nonautonomous nonlinear interconnected systems. An example is used to illustrate the application of the developed method.

Pullback random attractors of non-autonomous non-classical diffusion equations driven by colored noise

Pullback random attractors of non-autonomous non-classical diffusion equations driven by colored noise

Optimal decay rates for semi-linear non-autonomous evolution equations with vanishing damping

We consider a second order non-autonomous evolution equation in a Hilbert space Hu′′(t)+Au(t)+γ(t)u′(t)+∇F(u(t))=0,where A is a self-adjoint and nonnegative operator on H with a nontrivial kernel, the nonlinear term ∇F is the gradient of a differentiable function F:D(A1/2)→R, and γ(t) is a damping coefficient, vanishing as t goes to infinity. We focus on a limit case γ(t)=α/t (with some α>0), which turns out to be quite tricky to deal with. When α exceeds a critical value, we obtain the optimal decay rate of the solution of the equation, in terms of the exponent associated with F. Moreover, we present two examples to illustrate our theoretical results.

Positive almost periodic solutions of nonautonomous evolution equations and application to Lotka–Volterra systems

The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: . We assume that the family of closed linear operators on a Banach lattice satisfies the “Acquistapace–Terreni” conditions, so that the associated evolution family is positive and has an exponential dichotomy on . The nonlinear term , acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to , and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment.

Weak mean random attractors for nonautonomous stochastic parabolic equation with variable exponents

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.

CAUCHY PROBLEMS OF NONLINEAR NONAUTONOMOUS FRACTIONAL EVOLUTION EQUATIONS

This paper will focus on nonlinear nonautonomous abstract evolution equations with the Riemann–Liouville fractional derivative. The new concepts of classical and mild solutions will be introduced to carry out the study of solvability. Our methods are based on analytic semigroup theory, the probability density function, the fixed point technique and the compactness technique. The main results can be applied to the time variable fractional diffusion equations.

Nonautonomous Evolution Equation of Monge–Ampere Type with Two Space Variables

Nonautonomous Evolution Equation of Monge–Ampere Type with Two Space Variables

Input-to-state practical stability criteria of non-autonomous infinite-dimensional systems

We develop tools for the investigation of input-to-state practical stability (ISpS) and integral input-to-state practical stability (iISpS) of non-autonomous infinite-dimensional systems in Banach spaces. Sufficient conditions of ISpS and iISpS are given based on indefinite Lyapunov functions. The practical stability analysis is accomplished with the help of scalar practical stable functions. Then, a construction of ISpS Lyapunov function for a class of non-autonomous evolutions equations is provided in Hilbert spaces. We propose the ISpS Lyapunov methodology to make it suitable for the analysis of ISpS w.r.t. inputs from -spaces. Furthermore, we illustrate the theory with an example of a semi-linear reaction-diffusion equation.

Pullback trajectory attractor for nonautonomous wave equations

We consider the dynamical behavior for nonautonomous wave equations without uniqueness. The existence of both pullback weak and strong trajectories is proved. In order to obtain pullback strong trajectory attractor, we use the measure of noncompactness to verify evolutionary process is pullback strong asymptotically compact. The external force is not needed to be translation bounded.

Invariant measures and stochastic Liouville type theorem for non-autonomous stochastic reaction-diffusion equations

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

Approximation formula for a propagator in symmetrically normed ideals

We extend a result of Zagrebnov and prove an approximation formula for a propagator of an abstract Cauchy problem for non-autonomous evolution equation in the norm of arbitrary Banach ideal.

Stability and robust stability of non-autonomous linear differential equations with infinite delay

By the idea of comparison principle, we first introduce novel explicit criteria for the $ \varrho $-stability of general linear non-autonomous functional differential equations with infinite delay. Some characterizations for the time-independent exponential stability are also investigated. We then present a sufficient condition for the exponential stability of the equations subject to time-varying structured perturbations. These results lead to applications to the linear non-autonomous differential system with unbounded delay and the linear non-autonomous integro-differential equations with delay. Some examples are given to illustrate the obtained results.

Existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions

In this paper, we consider the existence and uniqueness of global solutions for non-autonomous evolution equations with state-dependent nonlocal conditions, in which the undelayed part admits an evolution operator. We discuss the problems by utilizing theory of evolution operators, Schauder fixed point theorem and Banach fixed point theorem. Some new results on existence and uniqueness of solutions of the considered equation are obtained on the infinite internal [0,+?). In the end, the obtained results are applied to a class of non-autonomous heat equations with state-dependent nonlocal conditions.

Determining modes and determining nodes for the 3D non-autonomous regularized magnetohydrodynamics equations

In this article, we investigate the asymptotic behavior of solutions for a non-autonomous regularized magnetohydrodynamics equations on 3D bounded domains. More precisely, the upper bounds on the number of determining modes and determining nodes for the system are established. The results show that the asymptotic behavior of the weak solution can be determined completely by its first finite number of Fourier modes and the large time behavior of the strong solution can be determined by its values on a finite number of points.

Attractors. Then and now

This survey is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds, as well as the dimension theory for the classes of attractors mentioned above. The theoretical results are illustrated by a number of clarifying examples and counterexamples. Bibliography: 248 titles.

Numerical dynamics of integrodifference equations: Forward dynamics and pullback attractors

In order to determine the dynamics of nonautonomous equations both their forward and pullback behavior need to be understood. For this reason we provide sufficient criteria for the existence of such attracting invariant sets in a general setting of nonautonomous difference equations in metric spaces. In addition it is shown that both forward and pullback attractors, as well as forward limit sets persist and that the latter two notions even converge under perturbation.As concrete application, we study integrodifference equations over the continuous functions under spatial discretization of collocation type. Integrodifference equation and Pullback attractor and Forward attractor and Urysohn operator

Attractors. Then and now

This survey is dedicated to the 100th anniversary of Mark Iosifovich Vishik and is based on a number of mini-courses taught by the author at the University of Surrey (UK) and Lanzhou University (China). It discusses the classical and modern results of the theory of attractors for dissipative PDEs, including attractors for autonomous and non-autonomous equations, dynamical systems in general topological spaces, various types of trajectory, pullback and random attractors, exponential attractors, determining functionals and inertial manifolds, as well as the dimension theory for the classes of attractors mentioned above. The theoretical results are illustrated by a number of clarifying examples and counterexamples. Bibliography: 248 titles.