Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

2000 ◽  
Vol 102 (4) ◽  
pp. 4339-4347
Author(s):  
Yu. B. Yanushanets
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solution ◽  
Anisotropic Media ◽  
Boltzman Equation ◽  
The Cauchy Problem ◽  
The One

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 Comparison between the Cauchy problem and the scattering problem for the Landau damping in the Vlasov-HMF equation

2021 ◽  
pp. 1-24
Author(s):  
Dario Benedetto ◽  
Emanuele Caglioti ◽  
Stefano Rossi
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Scattering Problem ◽  
New Method ◽  
Landau Damping ◽  
The Cauchy Problem ◽  
The One ◽  
Perturbative Result

We analyze the analytic Landau damping problem for the Vlasov-HMF equation, by fixing the asymptotic behavior of the solution. We use a new method for this “scattering problem”, closer to the one used for the Cauchy problem. In this way we are able to compare the two results, emphasizing the different influence of the plasma echoes in the two approaches. In particular, we prove a non-perturbative result for the scattering problem.

Classical solutions to a dissipative hyperbolic geometry flow in two space variables

2019 ◽  
Vol 16 (02) ◽  
pp. 223-243
Author(s):  
De-Xing Kong ◽  
Qi Liu ◽  
Chang-Ming Song
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Scalar Curvature ◽  
Hyperbolic Geometry ◽  
Existence Of Solutions ◽  
Classical Solutions ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
The One ◽  
Uniformly Bounded

We investigate a dissipative hyperbolic geometry flow in two space variables for which a new nonlinear wave equation is derived. Based on an energy method, the global existence of solutions to the dissipative hyperbolic geometry flow is established. Furthermore, the scalar curvature of the metric remains uniformly bounded. Moreover, under suitable assumptions, we establish the global existence of classical solutions to the Cauchy problem, and we show that the solution and its derivative decay to zero as the time tends to infinity. In addition, the scalar curvature of the solution metric converges to the one of the flat metric at an algebraic rate.

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}.

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

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