Oleksiy V. Kapustyan

On Global Attractors of Multivalued Semiprocesses and Nonautonomous Evolution Inclusions

In the first chapter, we considered the existence and properties of global attractors for autonomous multivalued dynamical systems. When the equation is nonautonomous, new and challenging difficulties appear. In this case, if uniqueness of the Cauchy problem holds, then the usual semigroup of operators becomes a two-parameter semigroup or process [38, 39], as we have to take into account the initial and the final time of the solutions.

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.

Strong attractors for vanishing viscosity approximations of non-Newtonian suspension flows

In this paper we prove the existence of global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of complex fluids. We also show that the attractors tend to the set of all complete bounded trajectories of the original problem when the parameter of the approximations goes to zero.

Properties of Resolving Operator for Nonautonomous Evolution Inclusions: Pullback Attractors

One of the most effective approaches to investigate nonlinear problems, represented by partial differential equations, inclusions and inequalities with boundary values, consists in the reduction of them into differential-operator inclusions, in infinite-dimensional spaces governed by nonlinear operators. In order to study these objects, the modern methods of nonlinear analysis have been used [7, 10, 11, 26]. Convergence of approximate solutions to an exact solution of the differential-operator equation or inclusion is frequently proved on the basis of the property of monotony or pseudomonotony of the corresponding operator.

Abstract Theory of Multivalued Semiflows

Beginning from the pioneering works [3, 52], the theory of global attractors of infinite-dimensional dynamical systems has become one of the main objects for investigation. Since then, deep results about existence, properties, structure, and dimension of global attractors for a wide class of dissipative systems have been obtained (see, e.g., [7, 38, 54, 75, 78]). For the application of this classical theory to partial and functional differential equations, it was necessary to have global existence and uniqueness of solutions of the Cauchy problem for all initial data of the phase space.

Attractors for Lattice Dynamical Systems

A lot of processes coming from Physics, Chemistry, Biology, Economy, and other sciences can be described using systems of reaction-diffusion equations. In this chapter, we study the asymptotic behavior of the solutions of a system of infinite ordinary differential equations (a lattice dynamical system) obtained after the spacial discretization of a system of reaction-diffusion equations in an unbounded domain. This kind of dynamical systems is then of importance in the numerical approximations of physical problems.

Auxiliary Properties of Evolution Inclusions Solutions for Earth Data Processing

A great number of collectives of mathematicians, mechanicians, geophysicists (mainly theorists), engineers goes in for qualitative investigation of nonlinear mathematical models of evolution processes and fields of different nature, in particular, problems deal with the dynamics of solutions of non-stationary problems. Far from complete list of results concern the given direction is in works [4, 5, 7, 9–17, 19].