Yashan B.O. Boundary Value Problems with Impulse Conditions for Parabolic Equations with Degeneration. - Qualifying scientific work on the rights of the manuscript.
The thesis is presented for the degree of Doctor of Philosophy in specialty 111- "Mathematics". - Yuri Fedkovych Chernivtsi National University, Chernivtsi, 2020.
The dissertation consists of an introduction, four chapters, conclusions and a list of references. In the introduction there is the relevance of the research topic, the purpose, task, subject, object and methods of research are formulated, scientific novelty, practical significance of the obtained results, connection of work with scientific topics and personal contribution of the applicant are indicated, as well as data about where the main results of the dissertation were reported, discussed and published is substantiated.
Chapter 1 provides an overview of the literature related to the topic of the dissertation.
In Chapter 2 problems for parabolic equations of the second order with impulsive action in time variable, namely: Cauchy and Dirichlet problem, problem with oblique derivative and unilateral boundary value problem are considered. The coefficients of the equation have power singularities of arbitrary order in time and space variables on a certain set of points. Using the principle of maximum and a priori estimates the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are proved.
In Chapter 3 the Cauchy problem, Dirichlet, problem with oblique derivative and unilateral boundary value problem with a multipoint condition in the time variable for a second-order parabolic equation with power degeneracy in the coefficients of the equation and boundary conditions are studied. The coefficients of the equation and the boundary condition allow power singularities of arbitrary order for any variables on a certain set of points. Necessary and sufficient conditions of existence of the solution are proved using the principle of maximum and a priori estimates in Hölder spaces with power weight. The order of power weight depends on the power singularities of the coefficients of the equation and boundary conditions.
In Chapter 4 the problem of optimal system control which is described by the problem with oblique derivative and integral condition of the time variable for the second order parabolic equation is investigated. Cases of internal, starting and boundary control are considered. The quality criterion is given by the sum of volume and surface integrals. The coefficients of the parabolic equation and the boundary condition allow power singularities of arbitrary order for any variables. Using the principle of maximum and a priori estimates existence and uniqueness of the solution of nonlocal parabolic boundary value problem with degeneration is established. Necessary and sufficient conditions for the existence of optimal solution of the system control problem which is described by the problem with oblique derivative for the parabolic equation with degeneracy are obtained.
The practical significance of the results.
The results of the dissertation have a theoretical nature. They can be used in further theoretical investigations of boundary value problems with impulse conditions for partial differential equations with degeneracies in coefficients.
Key words: interpolation inequalities, maximum principle, a priori estimates, degeneration, impulse action, boundary conditions.
