The fundamental matrix of solutions of the Cauchy problem for a class of parabolic systems of the Shilov type with variable coefficients

2011 ◽  
Vol 175 (4) ◽  
pp. 450-476 ◽  
Author(s):  
Vladyslav A. Litovchenko ◽  
Iryna M. Dovzhitska
Keyword(s):  
Cauchy Problem ◽  
Fundamental Matrix ◽  
Parabolic Systems ◽  
Variable Coefficients ◽  
The Cauchy Problem ◽  
Fundamental Matrix Of Solutions

scholarly journals Parabolic systems of Shilov-type with coefficients of bounded smoothness and nonnegative genus

2017 ◽  
Vol 9 (1) ◽  
pp. 72-85
Author(s):  
V.A. Litovchenko ◽  
G.M. Unguryan
Keyword(s):  
Cauchy Problem ◽  
Space Variable ◽  
Parabolic Systems ◽  
Type Systems ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Modern Theory ◽  
New Class ◽  
The Cauchy Problem ◽  
Derivatives Of

The Shilov-type parabolic systems are parabolically stable systems for changing its coefficients unlike of parabolic systems by Petrovskii. That's why the modern theory of the Cauchy problem for class by Shilov-type systems is developing abreast how the theory of the systems with constant or time-dependent coefficients alone. Building the theory of the Cauchy problem for systems with variable coefficients is actually today. A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with variable coefficients that includes a class of the Shilov-type systems with time-dependent coefficients and non-negative genus is considered in this work. A main part of differential expression concerning space variable $x$ of each such system is parabolic (by Shilov) expression. Coefficients of this expression are time-dependent, but coefficients of a group of younger members may depend also a space variable. We built the fundamental solution of the Cauchy problem for systems from this class by the method of sequential approximations. Conditions of  minimal smoothness on  coefficients of the systems by  variable $x$ are founded, the smoothness of solution is investigated and estimates of derivatives of this solution are obtained.  These results are important for  investigating  of the correct solution of the Cauchy problem for this systems in different functional spaces, obtaining forms of description of the solution of this problem and its properties.

scholarly journals Parabolic by Shilov systems with variable coefficients

2018 ◽  
Vol 9 (2) ◽  
pp. 145-153
Author(s):  
V.A. Litovchenko
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Systems ◽  
Variable Coefficients ◽  
Time Dependent ◽  
Transform Method ◽  
The Matrix ◽  
The Cauchy Problem ◽  
Matrix Symbol ◽  
The Fourier Transform

Because of the parabolic instability of the Shilov systems to change their coefficients, the definition parabolicity of Shilov for systems with time-dependent $t$ coefficients, unlike the definition parabolicity of Petrovsky, is formulated by imposing conditions on the matricant of corresponding dual by Fourier system. For parabolic systems by Petrovsky with time-dependent coefficients, these conditions are the property of a matricant, which follows directly from the definition of parabolicity. In connection with this, the question of the wealth of the class Shilov systems with time-dependent coefficients is important.A new class of linear parabolic systems with partial derivatives to the first order by the time $t$ with time-dependent coefficients is considered in this work. It covers the class by Petrovsky systems with time-dependent younger coefficients. A main part of differential expression of each such system is parabolic (by Shilov) expression with constant coefficients. The fundamental solution of the Cauchy problem for systems of this class is constructed by the Fourier transform method. Also proved their parabolicity by Shilov. Only the structure of the system and the conditions on the eigenvalues of the matrix symbol were used. First of all, this class characterizes the wealth by Shilov class of systems with time-dependents coefficients.Also it is given a general method for investigating a fundamental solution of the Cauchy problem for Shilov parabolic systems with positive genus, which is the development of the well-known method of Y.I. Zhitomirskii.

scholarly journals Peculiarities of the Fundamental Solution of Parabolic Systems with a Negative Genus

2020 ◽  
Author(s):  
Vladyslav Antonovich Litovchenko
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Classical Theory ◽  
Parabolic Systems ◽  
Type Systems ◽  
Variable Coefficients ◽  
The Cauchy Problem

For the parabolic Shilov-type systems with a negative genus, a method of studying the properties of a fundamental solution of the Cauchy problem is proposed. This method allows to improve the known estimates of Zhitomirskii fundamental solution for systems with dissipative parabolicity and describe the features of this solution more accurately. It opens wide possibilities for constructing a classical theory of the Cauchy problem for parabolic systems with negative genus and variable coefficients.

Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

2012 ◽  
Vol 10 (3) ◽  
pp. 1084-1102 ◽  
Author(s):  
Vladyslav A. Litovchenko ◽  
Iryna M. Dovzhytska
Keyword(s):  
Cauchy Problem ◽  
Parabolic Systems ◽  
Variable Coefficients
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