# 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

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

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.