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 strong maximum principle for parabolic differential equations

1986 ◽  
Vol 29 (1) ◽  
pp. 93-96 ◽  
Author(s):  
Wolfgang Walter
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Heat Equation ◽  
Nonlinear Equations ◽  
Strong Maximum Principle ◽  
Parabolic Differential Equations ◽  
Carry Over ◽  
Image Position

In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equationor to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.

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.

scholarly journals The maximum principle for a type of hereditary semilinear differential equation

1994 ◽  
Vol 36 (2) ◽  
pp. 194-212
Author(s):  
Feiyue He
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Optimal Control Problem ◽  
Control Problem ◽  
The Maximum Principle ◽  
Semilinear Differential Equation

AbstractAn optimal control problem governed by a class of delay semilinear differential equations is studied. The existence of an optimal control is proven, and the maximum principle and approximating schemes are found. As applications, three examples are discussed.

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

Sign in / Sign up

Export Citation Format

Share Document