Michael Z. Zgurovsky

Uniform Trajectory Attractors for Non-autonomous Nonlinear Systems

In this chapter we study uniform trajectory attractors for non-autonomous nonlinear systems. In Sect. 8.1 we establish the existence of uniform trajectory attractor for non-autonomous reaction-diffusion equations with Caratheodory’s nonlinearity. Section 8.2 devoted to structural properties of the uniform global attractor for non-autonomous reaction-diffusion system in which uniqueness of Cauchy problem is not guarantied. In the case of translation compact time-depended coefficients it is established that the uniform global attractor consists of bounded complete trajectories of corresponding multi-valued processes. Under additional sign conditions on non-linear term we also prove (and essentially use previous result) that the uniform global attractor is, in fact, bounded set in (L^{infty }(varOmega )cap H_0^1(varOmega )). Section 8.3 devoted to uniform trajectory attractors for nonautonomous dissipative dynamical systems. As applications we may consider FitzHugh–Nagumo system (signal transmission across axons), complex Ginzburg–Landau equation (theory of superconductivity), Lotka–Volterra system with diffusion (ecology models), Belousov–Zhabotinsky system (chemical dynamics) and many other reaction-diffusion type systems from Sect. 2.4.

Advances in the 3D Navier-Stokes Equations

In this chapter we provide a criterion for the existence of global strong solutions for the 3D Navier-Stokes system for any regular initial data. Moreover, we establish sufficient conditions for Leray-Hopf property of a weak solution for the 3D Navier-Stokes system. Under such conditions this weak solution is rightly continuous in the standard phase space H endowed with the strong convergence topology.

Qualitative Methods for Classes of Nonlinear Systems: Constructive Existence Results

In this chapter we establish the existence results for classes of nonlinear systems. Section 2.1 devoted to the first order differential-operator equations and inclusions. In Sect. 2.2 we consider the second order operator differential equations and inclusions in special classes of infinite-dimensional spaces of distributions. Section 2.3 devoted to the existence of strong solutions for evolutional variational inequalities with nonmonotone potential. The penalty method for strong solutions is justified. A nonlinear parabolic equations of divergent form are considered as examples of applications in Sect. 2.4.

Regularity of Solutions for Nonlinear Systems

In this chapter we establish sufficient conditions for regularity of all weak solutions for nonlinear systems. We note that the respective Cauchy problems may have nonunique weak solution. In Sect. 2.1 we establish regularity of all weak solutions for parabolic feedback control problems. Section 2.2 devoted to artificial control method for nonlinear partial differential equations and inclusions. The regularity of all weak solutions is obtained. In Sect. 2.3 we consider regularity results of all weak solutions for nonlinear reaction-diffusion systems with nonlinear growth. In Sect. 2.4 we consider the following examples of applications: a parabolic feedback control problem; a model of conduction of electrical impulses in nerve axons; a climate energy balance model; FitzHugh–Nagumo System; a model of combustion in porous media.

Strongest Convergence Results for Weak Solutions of Feedback Control Problems

In this chapter we establish strongest convergence results for weak solutions of feedback control problems. In Sect. 5.1 we set the problem. Section 5.2 devoted to the regularity of all weak solutions and their additional properties. In Sect. 5.3 we consider convergence of weak solutions results in the strongest topologies. As examples of applications we consider a model of combustion in porous media; a model of conduction of electrical impulses in nerve axons; and a climate energy balance model.

Strongest Convergence Results for Weak Solutions of Differential-Operator Equations and Inclusions

In this chapter we establish strongest convergence results for weak solutions of differential-operator equations and inclusions. In Sect. 6.1 we consider first order differential-operator equations and inclusions . Section 6.2 devoted to convergence results for weak solutions of second order operator differential equations and inclusions. In Sect. 6.3 we consider the following examples of applications: nonlinear parabolic equations of divergent form; nonlinear problems on manifolds with and without boundary: a climate energy balance model; a model of conduction of electrical impulses in nerve axons; viscoelastic problems with nonlinear “reaction-displacement” law.

Indirect Lyapunov Method for Autonomous Dynamical Systems

In this chapter we establish indirect Lyapunov method for autonomous dynamical systems. Section 9.1 devoted to the first order autonomous differential-operator equations and inclusions. In Sect. 9.2 we consider the second order autonomous operator differential equations and inclusions. In Sect. 9.3 we examine examples of applications. In particular, a model of combustion in porous media; a model of conduction of electrical impulses in nerve axons; viscoelastic problems with nonlinear “reaction-displacement” law etc.

Strongest Convergence Results for Weak Solutions of Non-autonomous Reaction-Diffusion Equations with Carathéodory’s Nonlinearity

In this chapter we consider the problem of uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion system with Caratheodory’s nonlinearity satisfying standard sign and polynomial growth assumptions. The main contributions of this chapter are: the uniform convergence results for all globally defined weak solutions of non-autonomous reaction-diffusion equations with Caratheodory’s nonlinearity and sufficient conditions for the convergence of weak solutions in strongest topologies.

Uniform global attractors for non-autonomous dissipative dynamical systems

In this paper we consider sufficient conditions for the existence of uniform compact global attractor for non-autonomous dynamical systems in special classes of infinite-dimensional phase spaces. The obtained generalizations allow us to avoid the restrictive compactness assumptions on the space of shifts of non-autonomous terms in particular evolution problems. The results are applied to several evolution inclusions.

Modeling and investigating the behavior of complex socio-economic systems

In the paper the problems of modeling and investigating the behavior of complex socio-economic systems are considered. Authors are deeply concerned about decision-making support issues in the case of evaluation for such complex socio-economic systems. To solve these problems, the totality of the development goals of the object and the priorities of these goals are determined, then the indicators are formed in to multifactor model that allows to assess the dynamics of achieving the objectives and the impact of the adoption of certain management decisions. This research includes the development of new approaches for system coherence of data of various nature, methods of aggregation and verification for data, adaptation of data mining methods for applied interdisciplinary research.