The fundamental solution and the Cauchy problem for parabolic systems

1964 ◽  
pp. 121-154
Author(s):  
L. N. Slobodeckiĭ
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Systems ◽  
The Cauchy Problem

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 Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

2003 ◽  
Vol 8 (1) ◽  
pp. 61-75
Author(s):  
V. Litovchenko
Keyword(s):  
Functional Analysis ◽  
Cauchy Problem ◽  
Fundamental Solution ◽  
Initial Function ◽  
Well Posedness ◽  
Fractional Differentiation ◽  
Basic Tool ◽  
Conjugate Space ◽  
Analysis Technique ◽  
The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

scholarly journals ON A NONLOCAL PROBLEM FOR PARTIAL DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

2020 ◽  
Vol 8 (2) ◽  
pp. 24-39
Author(s):  
V. Gorodetskiy ◽  
R. Kolisnyk ◽  
O. Martynyuk
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Parabolic Type ◽  
Generalized Functions ◽  
Initial Condition ◽  
Partial Derivatives ◽  
Time Problem ◽  
The Cauchy Problem ◽  
Space Of Generalized Functions ◽  
Derivatives Of

Spaces of $S$ type, introduced by I.Gelfand and G.Shilov, as well as spaces of type $S'$, topologically conjugate with them, are natural sets of the initial data of the Cauchy problem for broad classes of equations with partial derivatives of finite and infinite orders, in which the solutions are integer functions over spatial variables. Functions from spaces of $S$ type on the real axis together with all their derivatives at $|x|\to \infty$ decrease faster than $\exp\{-a|x|^{1/\alpha}\}$, $\alpha > 0$, $a > 0$, $x\in \mathbb{R}$. The paper investigates a nonlocal multipoint by time problem for equations with partial derivatives of parabolic type in the case when the initial condition is given in a certain space of generalized functions of the ultradistribution type ($S'$ type). Moreover, results close to the Cauchy problem known in theory for such equations with an initial condition in the corresponding spaces of generalized functions of $S'$ type were obtained. The properties of the fundamental solution of a nonlocal multipoint by time problem are investigated, the correct solvability of the problem is proved, the image of the solution in the form of a convolution of the fundamental solution with the initial generalized function, which is an element of the space of generalized functions of $S'$ type.

scholarly journals Fractional-hyperbolic systems

2013 ◽  
Vol 16 (4) ◽  
Author(s):  
Anatoly Kochubei
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Fractional Derivative ◽  
Hyperbolic Systems ◽  
Time Variable ◽  
Spatial Variables ◽  
First Order ◽  
Linear Partial Differential Equations ◽  
The Cauchy Problem ◽  
Evolution Systems

AbstractWe describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order α ∈ (0, 1) in the time variable t and the first order derivatives in spatial variables x = (x 1, …, x n), which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone {(t,x) : |t -α| ≤ 1}.

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