On the Kneser property for the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion

We study the Kneser property (i.e. the compactness and connectedness for the attainability set of solutions) for a reaction-diffusion system including as a particular case the complex Ginzburg-Landau equation and the Lotka-Volterra system with diffusion. Using this property we obtain also that the global attractor of this system in both the autonomous and non-autonomous cases is connected.     

Global Attractor of an Evolution Inclusion with Pulse Influence at Fixed Moments of Time

We consider an autonomous evolution inclusion with pulse perturbations at fixed moments of time. Under the conditions of global solvability, we prove the existence of a minimal compact set in the phase space that attracts all trajectories.     

Averaged Synthesis of the Optimal Control for a Wave Equation

For a wave equation, we determine an optimal control in the feedback form and prove the convergence of the constructed approximate control to the exact one.     

Singularities of Solutions of One Class of Equations of Continuum Mechanics

We analyze one class of families of integral equations and describe the dependence of the singularities of solutions of integral equations on the dimensions of the families of kernels of equations. On the basis of these results, we propose procedures for the construction of approximate solutions for a small parameter.     

On the solvability of one class of parameterized operator inclusions

We consider a class of parameterized operator inclusions with set-valued mappings of type. Sufficient conditions for the solvability of these inclusions are obtained and the dependence of the sets of their solutions on functional parameters is investigated. Examples that illustrate the results obtained are given. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1619–1630, December, 2008.     

Global Attractor for Impulsive Reaction-Diffusion Equation

In this paper, we consider a reaction-diffusion equation with nonsmooth nonlinearity whose solutions have impulse effects at fixed moments of time. We show how this object generates a nonautonomous multivalued dynamical system and prove the existence of a compact semiinvariant global attractor in the phase space. __________ Published in Neliniini Kolyvannya, Vol. 8, No. 3, pp. 319–328, July–September, 2005.     

Weak and strong attractors for the 3D Navier-Stokes system

We study in this paper the asymptotic behaviour of the weak solutions of the three-dimensional Navier-Stokes equations. On the one hand, using the weak topology of the usual phase space H (of square integrable divergence free functions) we prove the existence of a weak attractor in both autonomous and nonautonomous cases. On the other, we obtain a conditional result about the existence of the strong attractor, which is valid under an unproved hypothesis. Also, with this hypothesis we obtain continuous weak solutions with respect to the strong topology of H.     

Strong solutions and trajectory attractors to the thin-film equation with absorption

We prove local and global in time existence of non-negative weak solutions to the thin-film equation with absorption and obtain sufficient conditions for extra regularity of these solutions. Moreover, for the class of global strong solutions, we show existence of a trajectory attractor.     

Research technique of metamaterials with the controlled effective negative refractive index and their characteristics

A design of the two-beam hybrid microwave interferometer for measuring refractive indices of metamaterials is described. It is based on the branching of radiation using a waveguide directional coupler. Radiations of the interferometer waveguide arms are summed and the phase disbalance of the interferometer arms is counted on the waveguide measuring line. The frequency dispersion of the negative refractive indices of the metamaterial samples with a variable distance between its constituent plates is experimentally investigated. A possibility of controlling the refractive index of a metamaterial is demonstrated.     

Singularities of Solutions of Singular Integral Equations

From the point of view of singularities of differential mappings, we determine the dependence between the dimension of a family of singular equations and the types of singularities of their solutions.