# Degenerate parabolic equation

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, and the references given there.

How to Cite This Entry:
Degenerate parabolic equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Degenerate_parabolic_equation&oldid=46614
This article was adapted from an original article by A.M. Il'in (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article