Optimal Distributed Globally Bounded Control for Parabolic–Hyperbolic Equations with Nonlocal Boundary Conditions and a Linear Quality Criterion

For the problem of optimal control of a parabolic-hyperbolic process with nonlocal point boundary conditions, an explicit form of the solution is obtained in the form of formal series according to the system of eigenfunctions, which are generated by the spatial differential operator and boundary conditions. At the same time, the unequivocal solvability of the intermediate problems is established for each iteration. In addition, sufficient conditions for the convergence of the series are established, which determine the obtained formal solution of the optimal control problem, which justifies its correctness system of differential equations; Lyapunov exponents; fractal dimensio

Optimal Distributed Globally Bounded Control for Parabolic–Hyperbolic Equations with Nonlocal Boundary Conditions and a Linear Quality Criterion

For the problem of optimal control of a parabolic-hyperbolic process with nonlocal point boundary conditions, an explicit form of the solution is obtained in the form of formal series according to the system of eigenfunctions, which are generated by the spatial differential operator and boundary conditions. At the same time, the unequivocal solvability of the intermediate problems is established for each iteration. In addition, sufficient conditions for the convergence of the series are established, which determine the obtained formal solution of the optimal control problem, which justifies its correctness system of differential equations; Lyapunov exponents; fractal dimensio

Uniform attractors for vanishing viscosity approximations of non-autonomous complex flows

In this paper we prove the existence of uniform global attractors in the strong topology of the phase space for semiflows generated by vanishing viscosity approximations of some class of non-autonomous complex fluids

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 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

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

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

Якісні властивості та скінченновимірність з точністю до малого параметра слабких розв’язків кліматологічної моделі Будико–Селлерса

A qualitative analysis of the solutions behavior for the Budyko–Sellers energy balance climate model, considered on the Riemannian manifold without the boundary, is carried out. The global existence of the weak solution for the investigated problem with arbitrary initial data from the phase space is established. Solutions properties and regularity are studied. The Lyapunov function is found. The theorems on the existence of global and trajectory attractors for multi-valued semi-flow generated by all weak solutions of the problem are proved. The properties of attractors are studied. The relationship between attractors and the space of complete trajectories of the problem is established. The character of attraction of solutions to global and trajectory attractors and their structure are investigated. The finite-dimensionality up to a small parameter of the solutions dynamics for the problem is established.Проведен качественный анализ поведения решений климатологической модели энергетического баланса Будыко–Селлерса, рассмотренной на римановом многообразии без края. Установлено глобальное существование слабого решения исследуемой задачи с произвольными начальными данными с фазового пространства, изучены его свойства, регулярность. Найдено функцию Ляпунова. Доказаны теоремы существования глобального и траекторного аттракторов для многозначного полупотока, порожденного всеми слабыми решениями задачи. Изучены свойства аттракторов, установлена взаимосвязь между ними и пространством полных траекторий задачи. Исследованы характер притяжения решений к глобальному и траекторному аттракторам и их структура. Получена конечномерность с точностью до малого параметра динамики решений задачи.Проведено якісний аналіз поведінки розв’язків кліматологічної мо-делі енергетичного балансу Будико–Селлерса, розглянутої на рімановому ба-гатовиді без краю. Установлено глобальне існування слабкого розв’язку дослі-джуваної задачі з довільними початковими даними з фазового простору, вивчено його властивості, регулярність. Знайдено функцію Ляпунова. Доведено теореми існування глобального та траєкторного атракторів для багатозначного півпотоку, породженого всіма слабкими розв’язками задачі. Вивчено вла-стивості атракторів, установлено взаємозв’язок між ними та простором повних траєкторій задачі. Досліджено характер притягнення розв’язків до глобального та траєкторного атракторів та їх структуру. Отримано скінченновимірність з точністю до малого параметра динаміки розв’язків задачі

Effect of particle size of starting materials on the structure and properties of biogenic hydroxyapatite/glass composites

The work is devoted to investigation of porous glass-ceramic composite materials on the basis of biogenic hydroxyapatite and sodium borosilicate glass prepared from starting powders with different particle sizes (<50 µm and <160 µm). Starting hydroxyapatite/glass weight ratio was 1.0/0.46 and sintering temperature was ∼800 °C. Microstructural characterization of the surface and fracture of the samples revealed a decrease in sizes of grains and pores with decreasing the particle size of the precursor powder. However, porosity of the composites practically did not depend on the particle size and was equal to 32.5–33.0%. The same tendency was observed for the compression strength (66–67 MPa). However, investigation of structural-mechanical properties using an indentation method, where dominant load is applied to the surface layers of sample, showed up the effect of the particle size of the starting powder on the mechanical properties of the composites: the smaller particle size, the higher mechanical properties