Parabolic partial differential equation

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

$$ u _ {t} - \sum _ { i,j=1}^n a _ {ij} ( x, t) u _ {x _ {i} x _ {j} } - \sum _ { i=1}^ { n } a _ {i} ( x, t) u _ { x _ i } - a( x, t) u = f( x, t), $$

where $ \sum a _ {ij} \xi _ {i} \xi _ {j} $ is a positive-definite quadratic form. The variable $ t $ is singled out and plays the role of time. A typical example of a parabolic partial differential equation is the heat equation

$$ u _ {t} - \sum _ { i=1}^{ n } u _ {x _ {i} x _ {i} } = 0. $$

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 $ \phi _ {t} = F( \phi _ {x _ {i} x _ {j} } , \phi _ {x _ {i} } , \phi , x _ {i} , t ) $, where $ F $ is a function of variables $ ( u _ {ij} , u _ {i} , u , x _ {i} , t ) $ such that for a certain $ \epsilon > 0 $ one has $ \epsilon \| \lambda \| ^ {2} \leq \sum _ {i,j} F _ {u _ {ij} } \lambda _ {i} \lambda _ {j} $ on the domain under consideration.

A semi-linear partial differential equation of the second order, i.e. one of the form $ \sum _ {i,j = 1 } ^ {n} a _ {ij} ( \partial ^ {2} \phi ) / ( \partial x _ {i} \partial x _ {j} )= f ( x , \phi , \phi _ {x _ {i} } ) $, is said to be of parabolic type if $ \mathop{\rm det} ( a _ {ij} ) = 0 $ 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