scholarly journals Modified Cauchy Problem with Impulse Action for Parabolic Shilov Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Galina Unguryan
Keyword(s):  
Cauchy Problem ◽  
Approximate Solution ◽  
Parabolic Problems ◽  
Classical Solutions ◽  
Initial Condition ◽  
Correct Solvability ◽  
Impulse Action

For parabolic Shilov equations with continuous coefficients, the problem of finding classical solutions that satisfy a modified initial condition with generalized data such as the Gelfand and Shilov distributions is considered. This condition arises in the approximate solution of parabolic problems inverse in time. It linearly combines the meaning of the solution at the initial and some intermediate points in time. The conditions for the correct solvability of this problem are clarified and the formula for its solution is found. Using the results obtained, the corresponding problems with impulse action were solved.

Approximate solution of the Cauchy problem by means of polynomial operators

1973 ◽  
Vol 24 (6) ◽  
pp. 679-681
Author(s):  
S. A. Spasokukotskaya
Keyword(s):  
Cauchy Problem ◽  
Approximate Solution ◽  
The Cauchy Problem ◽  
Polynomial Operators

On the classical solutions for a Rosenau–Korteweg-deVries–Kawahara type equation

2021 ◽  
pp. 1-23
Author(s):  
Giuseppe Maria Coclite ◽  
Lorenzo di Ruvo
Keyword(s):  
Cauchy Problem ◽  
Small Amplitude ◽  
Type Equation ◽  
Discrete Systems ◽  
Finite Depth ◽  
Capillary Waves ◽  
Classical Solutions ◽  
Well Posedness ◽  
Kawahara Equation ◽  
The Cauchy Problem

The Rosenau–Korteweg-deVries–Kawahara equation describes the dynamics of dense discrete systems or small-amplitude gravity capillary waves on water of a finite depth. In this paper, we prove the well-posedness of the classical solutions for 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.

scholarly journals Existence of approximate solution to abstract nonlocal Cauchy problem

1992 ◽  
Vol 5 (4) ◽  
pp. 363-373 ◽  
Author(s):  
L. Byszewski
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Approximate Solution ◽  
Closed Subset ◽  
Nonlocal Condition ◽  
Right Hand ◽  
The Right ◽  
Nonlocal Cauchy Problem

The aim of the paper is to prove a theorem about the existence of an approximate solution to an abstract nonlinear nonlocal Cauchy problem in a Banach space. The right-hand side of the nonlocal condition belongs to a locally closed subset of a Banach space. The paper is a continuation of papers [1], [2] and generalizes some results from [3].

scholarly journals Life-Span of Classical Solutions to Hyperbolic Inverse Mean Curvature Flow

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zenggui Wang
Keyword(s):  
Cauchy Problem ◽  
Life Span ◽  
Hyperbolic System ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Classical Solutions ◽  
Quasilinear Hyperbolic System ◽  
Inverse Mean Curvature Flow ◽  
The Cauchy Problem

In this paper, we investigate the life-span of classical solutions to hyperbolic inverse mean curvature flow. Under the condition that the curve can be expressed in the form of a graph, we derive a hyperbolic Monge–Ampère equation which can be reduced to a quasilinear hyperbolic system in terms of Riemann invariants. By the theory on the local solution for the Cauchy problem of the quasilinear hyperbolic system, we discuss life-span of classical solutions to the Cauchy problem of hyperbolic inverse mean curvature.

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.

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