Uniqueness of the solution of the cauchy problem for systems of loaded linear equations

1985 ◽  
Vol 37 (1) ◽  
pp. 94-95 ◽  
Author(s):  
V. M. Borok
Keyword(s):  
Cauchy Problem ◽  
Linear Equations ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution

scholarly journals On the Cauchy problem for a degenerate parabolic differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 555-558
Author(s):  
Ahmed El-Fiky
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Parabolic Differential Equation ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Degenerate Parabolic

The aim of this work is to prove the existence and the uniqueness of the solution of a degenerate parabolic equation. This is done using H. Tanabe and P.E. Sobolevsldi theory.

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

scholarly journals UNIQUENESS IN THE CAUCHY PROBLEM FOR PARABOLIC EQUATIONS

2003 ◽  
Vol 46 (2) ◽  
pp. 329-340 ◽  
Author(s):  
Elisa Ferretti
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Growth Conditions ◽  
Parabolic Differential Equations ◽  
The Maximum Principle ◽  
The Cauchy Problem ◽  
Primary 35K15 ◽  
Uniqueness Of The Solution ◽  
Boot Strapping ◽  
Uniformly Bounded

AbstractWe discuss the problem of the uniqueness of the solution to the Cauchy problem for second-order, linear, uniformly parabolic differential equations. For most uniqueness theorems the solution must be uniformly bounded with respect to the time variable $t$, but some authors have shown an interest in relaxing the growth conditions in time.In 1997, Chung proved that, in the case of the heat equation, uniqueness holds under the restriction: $|u(x,t)|\leq C\exp[(a/t)^{\alpha}+a|x|^2]$, for some constants $C,a>0$, $0\lt\alpha\lt1$. The proof of Chung’s theorem is based on ultradistribution theory, in particular it relies heavily on the fact that the coefficients are constants and that the solution is smooth. Therefore, his method does not work for parabolic operators with arbitrary coefficients. In this paper we prove a uniqueness theorem for uniformly parabolic equations imposing the same growth condition as Chung on the solution $u(x,t)$. At the centre of the proof are the maximum principle, Gaussian-type estimates for short cylinders and a boot-strapping argument.AMS 2000 Mathematics subject classification: Primary 35K15

scholarly journals Semiclassical Spectral Series Localized on a Curve for the Gross–Pitaevskii Equation with a Nonlocal Interaction

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1289
Author(s):  
Anton E. Kulagin ◽  
Alexander V. Shapovalov ◽  
Andrey Y. Trifonov
Keyword(s):  
Cauchy Problem ◽  
Linear Equations ◽  
Time Dependent ◽  
Nonlocal Interaction ◽  
Pitaevskii Equation ◽  
Spectral Series ◽  
Semiclassical Asymptotics ◽  
Eigenvalues And Eigenfunctions ◽  
The Cauchy Problem ◽  
Interaction Term

We propose the approach to constructing semiclassical spectral series for the generalized multidimensional stationary Gross–Pitaevskii equation with a nonlocal interaction term. The eigenvalues and eigenfunctions semiclassically concentrated on a curve are obtained. The curve is described by the dynamic system of moments of solutions to the nonlocal Gross–Pitaevskii equation. We solve the eigenvalue problem for the nonlocal stationary Gross–Pitaevskii equation basing on the semiclassical asymptotics found for the Cauchy problem of the parametric family of linear equations associated with the time-dependent Gross–Pitaevskii equation in the space of extended dimension. The approach proposed uses symmetries of equations in the space of extended dimension.

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