scholarly journals THE VALUE OF THE SOLUTION OF ONE PROBLEM FOR THE PSEUDOPARABOLIC EQUATION IN THE CLASS OF GELS

2020 ◽  
Vol 70 (2) ◽  
pp. 77-83
Author(s):  
U.K. Koylyshov ◽  
◽  
A.Zh. Aldashova ◽  
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Fundamental Solution ◽  
A Priori Estimates ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Initial Condition ◽  
Pseudoparabolic Equation ◽  
Volume Potential ◽  
The Cauchy Problem

This article discusses the Cauchy problem for a pseudo-parabolic equation in three-dimensional space. The result can be generalized to - dimensional space. The Cauchy problem for equations of parabolic and elliptic types is well studied. For a pseudo-parabolic equation using the previously constructed fundamental solution, evaluating the fundamental solution and its derivatives. Applying the Fourier transform with respect to and the Laplace transform with, we first obtained a priori estimates for the potentials of the initial condition and the volume potential in Hölder spaces. Further, using these results, we have proved an estimate of the solution of the Cauchy problem for the pseudo-parabolic equation in Hölder classes. A detailed proof of the estimation of the potentials of the initial condition, the volume potential, and the solution of the Cauchy problem for the pseudoparabolic equation is given

scholarly journals REPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE

2021 ◽  
Vol 9 (1) ◽  
pp. 189-199
Author(s):  
H. Pasichnyk ◽  
S. Ivasyshen
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Integral Representation ◽  
Initial Condition ◽  
Kolmogorov Type ◽  
Ultraparabolic Equation ◽  
Volume Potential ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Strong Degeneration

The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could infinitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.

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 Global Existence of the Three Dimensional Heat-conductive Incompressible Viscous Fluids

2018 ◽  
Vol 179 ◽  
pp. 01001
Author(s):  
Jie Wu
Keyword(s):  
Cauchy Problem ◽  
Heat Conductivity ◽  
A Priori Estimates ◽  
Three Dimensional ◽  
Global Solutions ◽  
Local Solution ◽  
Viscous Fluids ◽  
Heat Conducting ◽  
The Cauchy Problem ◽  
Incompressible Viscous Fluids

In this paper, we consider the Cauchy problem of non-stationary motion of heatconducting incompressible viscous fluids in ℝ3. About the heat-conducting incompressible viscous fluids, there are many mathematical researchers study the variants systems when the viscosity and heat-conductivity coefficient are positive. For the heat-conductive system, it is difficulty to get the better regularity due to the gradient of velocity of fluid own the higher order term. It is hard to control it. In order to get its global solutions, we must obtain the a priori estimates at first, then using fixed point theorem, it need the mapping is contracted. We can get a local solution, then applying the criteria extension. We can extend the local solution to the global solutions. For the two dimensional case, the Gagliardo-Nirenberg interpolation inequality makes use of better than the three dimensional situation. Thus, our problem will become more difficulty to handle. In this paper, we assume the coefficient of viscosity is a constant and the coefficient of heat-conductivity satisfying some suitable conditions. We show that the Cauchy problem has a global-in-time strong solution (u,θ) on ℝ3 ×(0, ∞).

scholarly journals N-ORDER MOMENT FUNCTION FOR THE SOLUTION OF THE CAUCHY PROBLEM FOR DIFFUSION EQUATION WITH RANDOM COEFFICIENTS

2017 ◽  
Vol 17 (8) ◽  
pp. 5-20
Author(s):  
T.V. Besedina
Keyword(s):  
Cauchy Problem ◽  
Diffusion Equation ◽  
Three Dimensional ◽  
Random Coefficients ◽  
Moment Function ◽  
Order Moment ◽  
Random Initial Condition ◽  
Initial Condition ◽  
Dimensional Diffusion ◽  
The Cauchy Problem

Formula for n-order moment function for the solution of the Cauchy problem for three-dimensional diffusion equation with random coefficients and random initial condition is derived.

scholarly journals The Cauchy problem for inhomogeneous parabolic Shilov equations

2021 ◽  
Vol 13 (2) ◽  
pp. 475-484
Author(s):  
I.M. Dovzhytska
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fundamental Solution ◽  
Partial Solution ◽  
Spatial Variable ◽  
Original Equation ◽  
Correct Solvability ◽  
Volume Potential ◽  
The Cauchy Problem ◽  
Decreasing Functions

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

scholarly journals Asymptotics of the mild solution of awave equation in three-dimensional space driven by a general stochastic measure

2019 ◽  
pp. 12-17
Author(s):  
I. M. Bodnarchuk
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Mild Solution ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Absolute Value ◽  
Stochastic Measure ◽  
The Cauchy Problem ◽  
Three Dimensional Space ◽  
General Stochastic

We study the Cauchy problem for a wave equation in three-dimensional space driven by a general stochastic measure. Under some assumptions, we prove that the mild solution tends to zero almost surely as the absolute value of the spatial variable tends to infinity.

scholarly journals Analytical Solution of the Cauchy Problem for a Nonstationary Three-dimensional Model of the Filtration Theory

2021 ◽  
Vol 87 (1) ◽  
pp. 118-133
Author(s):  
Aigerim T. Rakhymova ◽  
Mars B. Gabbassov ◽  
Anvarjon A. Ahmedov
Keyword(s):  
Cauchy Problem ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Functional Space ◽  
Nonlinear Filtration ◽  
Three Dimensional Model ◽  
The Cauchy Problem ◽  
Filtration Law ◽  
Inhomogeneous Porous Medium ◽  
Three Dimensional Space

This paper is devoted to the study of the Cauchy problem for a system of differential equations describing the unsteady flow of a compressible fluid in a homogeneous and inhomogeneous porous medium with a general nonlinear filtration law in three-dimensional space. In the work using the methods of four-dimensional mathematics, a special four-dimensional space of space and time coordinates was developed, as well as a functional space of regular functions, and analytical conditions were obtained on the general form of the nonlinear filtration law for which the Cauchy problem has a solution.

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