Some stabilization theorems for solutions of the Cauchy problem for Shilov-parabolic systems in classes of generalized functions

1988 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
V. V. Gorodetskii
Keyword(s):  
Cauchy Problem ◽  
Parabolic Systems ◽  
Generalized Functions ◽  
The Cauchy Problem

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.

Uniqueness of a solution to the Cauchy problem for parabolic systems

2016 ◽  
Vol 93 (3) ◽  
pp. 316-317 ◽  
Author(s):  
E. A. Baderko ◽  
M. F. Cherepova
Keyword(s):  
Cauchy Problem ◽  
Parabolic Systems ◽  
The Cauchy Problem ◽  
Uniqueness Of A Solution

Uniqueness of the Solution of the Cauchy Problem for Parabolic Systems

2019 ◽  
Vol 55 (6) ◽  
pp. 806-814 ◽  
Author(s):  
E. A. Baderko ◽  
M. F. Cherepova
Keyword(s):  
Cauchy Problem ◽  
Parabolic Systems ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution
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