View source for Degenerate parabolic equation

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A partial differential equation

$$ 
F ( t, x, Du)  =  0,
$$

where the function  $  F( t, x, q) $
has the following property: For some even natural number  $  p $, 
all roots  $  \lambda $
of the polynomial

$$ 
\sum _ {\alpha : \ 
p \alpha _ {0} + \alpha _ {1} + \dots + \alpha _ {n} = m }

\frac{\partial  F ( t, x, Du) }{\partial  q _  \alpha  }

\lambda ^ {\alpha _ {0} } ( i \xi ) ^ {\alpha  ^  \prime  }
$$

have non-positive real parts for all real  $  \xi $
and, for certain  $  \xi \neq 0 $, 
$  t $, 
$  x $, 
and  $  Du $, 
$   \mathop{\rm Re}  \lambda = 0 $
for some root  $  \lambda $, 
or for certain  $  t $, 
$  x $
and  $  Du $
the leading coefficient at  $  \lambda  ^ {m/p} $
vanishes. Here  $  t $
is an independent variable which is often interpreted as time;  $  x $
is an  $  n $-
dimensional vector  $  ( x _ {1} \dots x _ {n} ) $; 
$  u ( t, x) $
is the unknown function;  $  \alpha $
is a multi-index  $  ( \alpha _ {0} \dots \alpha _ {n} ) $; 
$  Du $
is the vector with components

$$ 
D  ^  \alpha  u  = \ 

\frac{\partial  ^ {| \alpha | } u }{\partial  t ^ {\alpha _ {0} }
\partial  x _ {1} ^ {\alpha _ {1} } \dots
\partial  x _ {n} ^ {\alpha _ {n} } }
 ,\ \ 
p \alpha _ {0} +
\sum _ {i= 1 } ^ { n }  \alpha _ {i} \leq  m ,
$$

$  q $
is a vector with components  $  q _  \alpha  $, 
$  \xi $
is an  $  n $-
dimensional vector  $  ( \xi _ {1} \dots \xi _ {n} ) $, 
and  $  ( i \xi ) ^ {\alpha  ^  \prime  } = ( i \xi _ {1} ) ^ {\alpha _ {1} } \dots ( i \xi _ {n} ) ^ {\alpha _ {n} } $. 
See also [[Degenerate partial differential equation|Degenerate partial differential equation]], and the references given there.