# Litovchenko V. A.

### Hyperbolic systems in Gelfand and Shilov spaces

↓ Abstract

Ukr. Mat. Zh. - 2019. - 71, № 10. - pp. 1360-1373

UDC 517.956.32, 517.955.2

For systems hyperbolic in Shilov's sense with time-dependent coefficients, the properties of the Green function are studied in the $S$-type spaces.
For systems of this kind in the indicated spaces, we establish the correct solvability of the Cauchy problem.
It is shown that, for each $\beta>1,$ the space ${S_0^\beta}'$ of Gelfand and Shilov distributions is the class of well-posedness of this problem.

### One method for the investigation of the fundamental solution of the Cauchy problem for parabolic systems

Ukr. Mat. Zh. - 2018. - 70, № 6. - pp. 801-811

A recursive method for the investigation of the fundamental solution of the Cauchy problem for parabolic Shilov systems with time-dependent coefficients is proposed. It is based on the general formula for the solution of linear inhomogeneous systems of differential equations of the first order and does not require the use of the genus of the analyzed system.

### Fundamental solution of the Cauchy problem for the Shilov-type parabolic systems with coefficients of bounded smoothness

Litovchenko V. A., Unguryan G.M.

Ukr. Mat. Zh. - 2017. - 69, № 3. - pp. 348-364

Under the condition of minimal smoothness of the coefficients, we construct the fundamental solution of the Cauchy problem and study the principal properties of this solution for a special class of linear parabolic systems with bounded variable coefficients covering the class of Shilov-type parabolic systems of nonnegative kind.

### Analogs of *S*-type spaces of partially even functions

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 512-521

We construct analogs of $S$-type spaces whose elements are functions that are even in a part of components of their arguments. We obtain a formula that expresses a power of a Bessel operator via the corresponding powers of a differential operator. This formula enables us to establish a relation between these spaces in terms of the Fourier – Bessel transformation and to clarify some basic properties of typical operations on their elements.

### Principle of localization of solutions of the Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type

Litovchenko V. A., Strybko O. V.

Ukr. Mat. Zh. - 2010. - 62, № 11. - pp. 1473–1489

In the case where initial data are generalized functions of the Gevrey-distribution type for which the classical notion of equality of two functions on a set is well defined, we establish the principle of local strengthening of the convergence of a solution of the Cauchy problem to its limit value as $t → +0$ for one class of degenerate parabolic equations of the Kolmogorov type with $\overrightarrow{2b}-$parabolic part whose coefficients are continuous functions that depend only on $t$.

### Cauchy problem for a class of degenerate kolmogorov-type parabolic equations with nonpositive genus

Ivasyshen S. D., Litovchenko V. A.

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1330–1350

We study the properties of the fundamental solution and establish the correct solvability of the Cauchy problem for a class of degenerate Kolmogorov-type equations with $\{\overrightarrow{p},\overrightarrow{h}\}$-parabolic part with respect to the main group of variables and nonpositive vector genus in the case where the solutions are infinitely differentiable functions and their initial values are generalized functions in the form of Gevrey ultradistributions.

### Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

Ivasyshen S. D., Litovchenko V. A.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1066-1087

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with \( \left\{ {\overrightarrow p, \overrightarrow h } \right\} \)-parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.

### Cauchy problem for one class of pseudodifferential systems with entire analytic symbols

Ukr. Mat. Zh. - 2006. - 58, № 9. - pp. 1211–1233

Using functions convex downward, we describe a class of pseudodifferential systems with entire analytic symbols that contains Éidel’man parabolic systems of partial differential equations with continuous time-dependent coefficients. We prove a theorem on the correct solvability of the Cauchy problem for these systems in the case where initial data are generalized functions. We also establish the principle of localization of a solution of this problem.

### Cauchy problem with Riesz operator of fractional differentiation

Ukr. Mat. Zh. - 2005. - 57, № 12. - pp. 1653–1667

In the class of generalized functions of finite order, we establish the correct solvability of the Cauchy problem for a pseudodifferential equation whose symbols are homogeneous functions of order γ > 0. We prove a theorem on the localization property of a solution of this problem.

### Correct Solvability of the Cauchy Problem for One Equation of Integral Form

Ukr. Mat. Zh. - 2004. - 56, № 2. - pp. 185-197

We describe spaces of test functions that generalize *S*-type and *W*-type spaces. In these spaces, we establish the complete solvability of the Cauchy problem for one equation of integral form with Bessel fractional integro-differential operator.

### Complete Solvability of the Cauchy Problem for Petrovskii Parabolic Equations in *S*-Type Spaces

Ukr. Mat. Zh. - 2002. - 54, № 11. - pp. 1467-1479

We establish the correct solvability (in both directions) of the Cauchy problem for Petrovskii parabolic equations with time-dependent coefficients in *S*-type spaces. We also prove that a solution of this problem stabilizes to zero in the sense of the topology of these spaces.

### On the Correct Solvability of One Cauchy Problem

Ukr. Mat. Zh. - 2002. - 54, № 8. - pp. 1067-1076

We establish a criterion for convolutors in certain *S*-type spaces. Using this criterion, we prove the correct solvability (in both directions) of one Cauchy problem in these spaces.