Uniqueness of the solution of the cauchy problem for some differential-difference parabolic equations

2004 ◽  
Vol 40 (10) ◽  
pp. 1461-1466 ◽  
Author(s):  
A. B. Muravnik
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
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 The Cauchy Problem for Parabolic Equations with Degeneration

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Ivan Pukal’skii ◽  
Bohdan Yashan
Keyword(s):  
Boundary Condition ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Arbitrary Order ◽  
Power Weight ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Set Of Points

Annotation. For a second-order parabolic equation, the multipoint in time Cauchy problem is considered. The coefficients of the equation and the boundary condition have power singularities of arbitrary order in time and space variables on a certain set of points. Conditions for the existence and uniqueness of the solution of the problem in Hölder spaces with power weight are found.

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

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