Pavlo O. Kasyanov

Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications

Abstract In this note we provide sufficient conditions for the existence of a Lyapunov function for a class of parabolic feedback control problems. The results are applied to the long-time behavior of state functions for the following problems: (i) a model of combustion in porous media; (ii) a model of conduction of electrical impulses in nerve axons; and (iii) a climate energy balance model.

Lyapunov Functions for Differential Inclusions and Applications in Physics, Biology, and Climatology

We investigate additional regularity properties of all globally defined weak solutions, their global and trajectory attractors for classes of semi-linear parabolic differential inclusions with initial data from the natural phase space. The main contributions in this note are: (i) sufficient conditions for the existence of a Lyapunov function for a class of parabolic feedback control problems; (ii) convergence results for all weak solutions in the strongest topologies; and (iii) new structure and regularity properties for global and trajectory attractors. Results applied to the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; and (c) a climate energy balance model.

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

Abstract In this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.

Long-Time Behavior of State Functions for Badyko Models

In this note we examine the long-time behavior of state functions for a climate energy balance model (Budyko Model) in the strongest topologies of the phase and the extended phase spaces. Strongest convergence results for all weak solutions are obtained. New structure and regularity properties for global and trajectory attractors are justified.

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.