Information on the thesis

Thesis for the Degree of Doctor of Sciences in Physics and Mathemati-
cs on the speciality 01.01.02  Dierential equations (111 Mathematic).
 Lviv Polytechnic National University, Pidstryhach Institute for Applied
Problems of Mechanics and Mathematics of NAS of Ukraine.  Lviv Ivan
Franko National University, Ministry of Education and Science of Ukraine,
Lviv, 2021.
The dissertation is devoted to construction, research and application
of fundamental solutions of the Cauchy problem for degenerate parabolic
equations from classes K 1 , K 2 , K 3 and K 4 . Class K 1 are ultraparabolic
equations of the Kolmogorov type. The class K 2 includes equations of the
Kolmogorov type of arbitrary order. An equation of the class K 3  is an
equation of the type of equations from the class K 1 , in which degenerations
are additionally present at t = 0. The classes of equation K 1 , K 2 and
K 3 is a natural generalization in dierent directions of the known classical
Kolmogorov’s equation of diusion with inertia. The class K 4 consists of
−→
2b
parabolic in the sense of Eidelman systems of equations and degeneration
on the initial hyperplane. The feature of the equations from this class is the
inequality of spatial variables and the presence of degeneracy on the initial
hyperplane. For equations from classes K 1 , K 2 and K 3 the conditions for
the coecients of the equations are found, according to which, with the
help of the stepwise Levy method, the classical fundamental solutions of
the Cauchy problem are constructed and investigated, and the estimates of
the constructed solutions and their derivatives are established, theorems on
correct solvability and integral images of solutions in families of L p -weight
spaces are proved, which have exponential growth of maximum order 2 or
2b under the condition |x| → ∞ according to the type dependent on t. For a
subclass of K 1  equations with real coecients are established, in addition
to the existence and estimates of the constructed fundamental solution Z,
additional properties Z (nonnegativity, normalization, convolution formula
etc.), which allow to interpret the function Z as the density of transition
probabilities of some diusion process; the integral representation is proved
and the correct solvability of the Cauchy problem in the class of nonnegative
functions is proved; formulas for determining the characteristics of such a
diusion process are obtained. The theorems on the local solvability of the
Cauchy problem for the corresponding quasilinear equation are also proved
and the existence of a global solution of the Cauchy problem for a semilinear
equation from the class is established K 1 .
The information obtained in the dissertation on the fundamental soluti-
ons of equations from classes K 1 , K 2 and K 3 in a certain way show how
the properties of singularities and degeneracies aect the properties of the
constructed fundamental solutions and the results of their applications (de-generacy of the matrix of coecients which are at the senior derivatives in
the equation, inequality of spatial variables, degeneracy under the condition
t = 0).
For equations from the class K 4 a comprehensive study of potentials is
carried out, the core of which is the corresponding fundamental solution in
the weight spaces of Holder functions, which correctly and accurately take
into account the behavior under the condition t → 0 of fundamental soluti-
on; the theorem on correct solvability, a priori estimates and increase of
smoothness of solutions of the Cauchy problem and local solvability of the
problem with degeneracy on the initial hyperplane is proved. All possible
types of degeneracy of equations under the condition t = 0 are considered.
The obtained results summarize and supplement the results previously obtai-
ned by the author for 2b- parabolic by Petrovsky systems of equations and
degeneration on the initial hyperplane .
The study which are conducted in the research have theoretical nature.
Its results and methods of obtaining them can be used to study analytical
methods of degenerate parabolic equations of more general structure, that
is, to construct and study fundamental solutions and their applications to
establish the correct solvability, integral representation, and properties of
solutions of the Cauchy problem for such equations.
Keywords: degenerate parabolic equations of Kolmogorov type, parabolic
equations with degenerations on the initial hyperplane, the Cauchy problem,
fundamental solution of the Cauchy problem, Levi’s method, volume potenti-
al, correct and local solvability.