# Parabolic partial differential equation

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

#### Comments

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.

#### References

[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: http://www.encyclopediaofmath.org/index.php?title=Parabolic_partial_differential_equation&oldid=28257