Information on the thesis

The thesis deals with the research of the problems in bounded cylindrical
domains with local multipoint conditions on time variable and certain conditions
in the other variables (conditions of periodicity, conditions of Dirichlet type) for
linear parabolic equations and systems of equations with variable coefficients.
In general case these problems are conditionally correct and their solvability is
related to the problem of small denominators.
The following new results have been obtained:
— the correctness conditions of two-point problem for Petrovskii parabolic
equations of second order on time with Sturm-Liouville operators on spatial19
coordinates are established; it is proved that these conditions are fulfilled for
an arbitrary fixed interpolation nodes and almost for all (with respect to the
Hausdorff measure) vectors composed of equation coefficients;
— the unique solvability of multipoint problem with simple nodes for Petro-
vskii parabolic equations of a high order with Sturm-Liouville operators on
spatial coordinates and for 2
~
b-parabolic equations with Bessel operator on one
of the spatial variables is established for almost all (with respect to Lebesgue
measure) vectors, which consist of interpolation nodes;
— the conditions of correct solvability of multipoint problems for Petrovskii
parabolic equations with variables on spatial coordinates of coefficients, and also
for factorized parabolic operator with coefficients depended on time and spatial
variables are established; metric theorems about lower-bound estimate of small
denominators to these problems are proved;
— the conditions of correct solvability of two-point problem for Petrovskii
parabolic systems of second order on time and problems with multipoint conditi-
ons for the systems of high order are established; it is proved that this conditions
are performed for almost all (with respect to the Lebesgue measure) vectors
composed of coefficients of systems, coefficients of multipoint conditions and
values of interpolation nodes;
— explicit formulas for solutions in form of Fourier series based on systems
of orthogonal functions of considered problems are constructed.
The thesis results are of theoretical importance. They can be applied in
further researches of problems with multipoint conditions on time for linear and
nonlinear parabolic equations and systems of equations, and also in the study of
specific problems of practice which are modelled by considered problems.
Key words: parabolic equations, multipoint conditions, divided differences,
Green’s function, small denominators, Lebesgue measure, Hausdorff measure.