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''subcaloric function''

The analogue of a [[Subharmonic function|subharmonic function]] for the heat equation

$$ \tag{* }

\frac{\partial  u }{\partial  t }
 - \Delta  ^ {2} u  =  0,
$$

where  $  u= u( x, t) $, 
$  x = ( x _ {1} \dots x _ {n} ) \in \mathbf R  ^ {n} $
and  $  \Delta  ^ {2} u = \sum _ {j=} 1  ^ {n} \partial   ^ {2} u / \partial  x _ {j}  ^ {2} $
is the [[Laplace operator|Laplace operator]]. For example, a function  $  v = v( x, t) $, 
$  x \in \mathbf R $, 
$  t > 0 $, 
of class  $  C  ^ {2} $
will be a subparabolic function in the rectangle

$$ 
D  =  \{ {( x, t) \in \mathbf R \times \mathbf R _ {+} } : {a < x < b, 0 < t < h } \}
$$

if

$$ 

\frac{\partial  v }{\partial  t }
 - 
\frac{\partial   ^ {2} v }{\partial  x  ^ {2} }
  \leq   0
$$

everywhere in  $  D $. 
In a more general case, let the point  $  ( x _ {0} , t _ {0} ) \in D $, 
let  $  \Delta $
be a sufficiently small equilateral triangle with base parallel to the axis  $  t= 0 $
and let  $  ( x _ {0} , t _ {0} ) \in \Delta \subset  D $. 
A function  $  v = v( x, t) $
that is continuous in the closed domain  $  \overline{D}\; $
is said to be subparabolic in  $  D $
if its value at any point  $  ( x _ {0} , t _ {0} ) \in D $
is not greater than the value at this point of that solution of (*) in any sufficiently small triangle  $  \Delta $, 
$  ( x _ {0} , t _ {0} ) \in \Delta $, 
that has the same values on the sides of  $  \Delta $
as  $  v( x, t) $.

Many properties of subharmonic functions, including the maximum principle, are also valid for subparabolic functions.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,   "A course of higher mathematics" , '''2''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.G. Petrovskii,   "Partial differential equations" , Saunders  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,   "Zur ersten Randwertaufgabe der Wärmeleitungsgleichung"  ''Compos. Math.'' , '''1'''  (1935)  pp. 383–419</TD></TR></table>

====Comments====
See [[#References|[a1]]] for an account of subparabolic functions from a potential-theoretic point of view.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.L. Doob,   "Classical potential theory and its probabilistic counterpart" , Springer  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Friedman,   "Partial differential equations of parabolic type" , Prentice-Hall  (1964)</TD></TR></table>