# 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