View source for Inverse parabolic partial differential equation

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An equation of the form

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

$$ 
= \ 
f ( x , t ) ,
$$

where the form  $  \sum a _ {ij} \xi _ {i} \xi _ {j} $
is positive definite. The variable  $  t $
plays the role of  "inverse"  time. The substitution  $  t = - t  ^  \prime  $
reduces equation (*) to the usual parabolic form. Parabolic equations of  "mixed"  type occur, for example,  $  u _ {t} = x u _ {xx} $
is a direct parabolic equation for  $  x > 0 $
and an inverse parabolic equation for  $  x < 0 $, 
with degeneracy of the order for  $  x = 0 $.

====Comments====
The [[Cauchy problem|Cauchy problem]] for an equation (*) is a well-known example of an ill-posed problem (cf. [[Ill-posed problems|Ill-posed problems]]). For a discussion of the backward heat equation (cf. also [[Thermal-conductance equation|Thermal-conductance equation]])

$$ 
u _ {t} + \Delta u  =  0
$$

( $  \Delta $
being the [[Laplace operator|Laplace operator]]) see [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. Payne,   "Improperly posed problems in partial differential equations" , SIAM  (1975)</TD></TR></table>