scholarly journals On the Kernel of a Polynomial of Scalar Derivations

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 515 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Cauchy Problem ◽  
Problem Solution ◽  
Cauchy Problems ◽  
Vector Variable ◽  
Gradient Representation

In this paper, by using a vector variable, the procedure of characteristic systems allows us to describe the kernel of a polynomial of scalar derivations by solving Cauchy Problems for the corresponding system of ODEs. Moreover, a gradient representation for the associated Cauchy Problem solution is derived.

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.

scholarly journals The behavior of solutions of non-linear differential equations in Hilbert space. I

1988 ◽  
Vol 11 (1) ◽  
pp. 143-165 ◽  
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Case ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Degenerate Linear

This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi-uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon-Nirenberg is used to study the uniqueness of the Cauchy problem for the non-degenerate linear case.

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