View source for Parabolic partial differential equation

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An equation (cf. [[Differential equation, partial|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|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 [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)  {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N.V. Krylov,   "Nonlinear elliptic and parabolic equations of the second order" , Reidel  (1987)  (Translated from Russian)  {{MR|0901759}} {{ZBL|0619.35004}} </TD></TR></table>