The maximum principle and local Lipschitz estimates near the lateral boundary for solutions of second-order parabolic equations

1976 ◽  
Vol 16 (6) ◽  
pp. 897-909
Author(s):  
L. I. Kamynin ◽  
B. N. Khimchenko
Keyword(s):  
Maximum Principle ◽  
Parabolic Equations ◽  
Second Order ◽  
Lateral Boundary ◽  
The Maximum Principle ◽  
Lipschitz Estimates ◽  
Local Lipschitz Estimates

scholarly journals Unique Continuation Property of Non-Positive Weak Subsolutions for Parabolic Equations of Higher Order

1964 ◽  
Vol 24 ◽  
pp. 241-248
Author(s):  
Kazunari Hayashida
Keyword(s):  
Maximum Principle ◽  
Differential Operator ◽  
Parabolic Equations ◽  
Unique Continuation ◽  
Second Order ◽  
Wide Sense ◽  
The Maximum Principle ◽  
Unique Continuation Property ◽  
Continuation Property ◽  
Order Continuous

When L is a parabolic differential operator of second order, Nirenberg [6] proved the maximum principle for the function u which has second order continuous derivatives and satisfies Lu≧0. Recently Friedman [2] has proved the maximum principle for the measurable function satisfying Lu≧O in the wide sense. This function is named a weakly L-subparabolic function. On the other hand, Littman [5] earlier than Friedman, has defined a weakly A- subharmonic function for an elliptic differential operator A of second order and has showed the maximum principle for it.

Phragmèn-Lindelöf and Comparison Theorems for Elliptic-Parabolic Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 864-871
Author(s):  
J. K. Oddson
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Comparison Theorems ◽  
Parabolic Differential Equations ◽  
The Maximum Principle ◽  
Comparison Functions

Theorems of Phragmèn-Lindelöf type and other related results for solutions of elliptic-parabolic equations have been given by numerous authors in recent years. Many of these results are based upon the maximum principle and the use of auxiliary comparison functions which are constructed as supersolutions of the equations under various conditions on the coefficients. In this paper we present an axiomatized treatment of these topics, replacing specific hypotheses on the nature of the coefficients of the equations by a single assumption concerning the maximum principle and another concerning the existence of positive supersolutions, in terms of which the theorems are stated.

scholarly journals On the maximum principle for a class of nonlinear parabolicequations

2017 ◽  
Vol 21 (6) ◽  
pp. 89-92
Author(s):  
A.A. Kon’kov
Keyword(s):  
Maximum Principle ◽  
Nonlinear Equations ◽  
Wide Class ◽  
Lower Order ◽  
Parabolic Equations ◽  
Linear Equations ◽  
Nonlinear Parabolic Equations ◽  
Spatial Variables ◽  
The Maximum Principle ◽  
Nonlinear Parabolic

In this paper, we consider solutions of nonlinear parabolic equations in the half-space.It is well-known that, in the case of linear equations, one needs to impose additional conditions on solutions for the validity of the maximum principle. The most famous of them are the conditions of Tikhonov and T¨acklind. We show that such restrictions are not needed for a wide class of nonlinear equations. In so doing, the coefficients of lower-order derivatives can grow arbitrarily as the spatial variables tend to infinity.We give an example which demonstrates an application of the obtained re- sults for nonlinearities of the Emden - Fowler type.

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