The Cauchy problem for some classes of quasi-linear parabolic equations

1969 ◽  
Vol 6 (3) ◽  
pp. 634-637
Author(s):  
S. N. Kruzhkov
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Linear Parabolic Equations ◽  
The Cauchy Problem

scholarly journals The Cauchy Problem of Linear Parabolic Equations with Discontinuous and Unbounded Coefficients

1971 ◽  
Vol 41 ◽  
pp. 33-42 ◽  
Author(s):  
Yoshiaki Ikeda
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Parabolic Equations ◽  
Unbounded Coefficients ◽  
Linear Parabolic Equations ◽  
The Cauchy Problem

In this article we shall prove the uniqueness and existense of a weak solution for the Cauchy problem of linear parabolic equations with discontinuous and unbounded coefficients(1.1)

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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