Nataliia V. Gorban

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.

On Regularity of All Weak Solutions and Their Attractors for Reaction-Diffusion Inclusion in Unbounded Domain

We consider the reaction-diffusion equation with multivalued function of interaction in an unbounded domain. Conditions on the parameters of the problem can not guarantee the uniqueness of the solution of the Cauchy problem. In this work we focus on the study of long-term forecasts of the state functions of reaction-diffusion equation with use of the theory of global attractors for multivalued semiflows. It is obtained the results of the existence and properties of all weak solutions. We obtain the standard a priori estimates for weak solutions of the investigated problem, prove the existence of weak solutions, the existence of global and trajectory attractors for the problem in phase and extended phase spaces respectively. We provide the regularity properties for all globally defined weak solutions and their global and trajectory attractors. The results can be used for the investigation of specific physical models including combustion models in porous media, conduction models of electrical impulses into the nerve endings, climate models.

On Global Attractors for Autonomous Damped Wave Equation with Discontinuous Nonlinearity

We consider autonomous damped wave equation with discontinuous nonlinearity. The long-term prognosis of the state functions when the conditions on the parameters of the problem do not guarantee uniqueness of solution of the corresponding Cauchy problem are studied. We prove the existence of a global attractor and investigate its structure. It is obtained that trajectory of every weak solution defined on \([0;+\infty )\) tends to a fixed point.

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 Global Attractor for Nonautonomous Reaction–Diffusion Equations with Carathéodory’s Nonlinearity

We consider nonautonomous reaction–diffusion system with Caratheodory’s nonlinearity. We investigate the long-time dynamics of all globally defined weak solutions under the standard sign and polynomial growth conditions. We obtain new topological properties of solutions, in particular flattening property, prove the existence of uniform global attractor for multivalued semiflow generated by considered problem.