Parabolic partial differential equation

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An equation (cf. Differential equation, partial) of the form

where is a positive-definite quadratic form. The variable is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the heat equation


The above defines second-order linear parabolic differential equations. There also exist notions of non-linear parabolic equations. For instance, in [a2] equations are studied of the form , where is a function of variables such that for a certain one has on the domain under consideration.

A semi-linear partial differential equation of the second order, i.e. one of the form , is said to be of parabolic type if at each point of the domain under consideration.


[a1] A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) MR0181836 Zbl 0144.34903
[a2] N.V. Krylov, "Nonlinear elliptic and parabolic equations of the second order" , Reidel (1987) (Translated from Russian) MR0901759 Zbl 0619.35004
How to Cite This Entry:
Parabolic partial differential equation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.P. Soldatov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article