Degenerate hyperbolic equation

A partial differential equation

$$ \tag{* } F ( t, x, Du ) = 0 $$

where the function $ F( t, x, q) $ satisfies the following condition: The roots of the polynomial

$$ \sum _ {| \alpha | = m } \frac{\partial F ( t, x, Du ) }{\partial q _ \alpha } \lambda ^ {\alpha _ {0} } \xi ^ {\alpha ^ \prime } $$

are real for all real $ \xi $, and there exist $ \xi \neq 0 $, $ t $, $ x $, and $ Du $ for which some of the roots either coincide or the coefficient of $ \lambda ^ {m} $ 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 $ and $ \alpha ^ \prime $ are multi-indices, $ \alpha = ( \alpha _ {0} \dots \alpha _ {n} ) $, $ \alpha ^ \prime = ( \alpha _ {1} \dots \alpha _ {n} ) $; $ Du $ is a vector with components

$$ D ^ \alpha u = \ \frac{\partial ^ {| \alpha | } u }{\partial t ^ {\alpha _ {0} } \partial x _ {1} ^ {\alpha _ {1} } \dots \partial x _ {n} ^ {\alpha _ {n} } } ; $$

only derivatives of an order not exceeding $ m $ enter in equation (*); the $ q _ \alpha $ are the components of a vector $ q $; $ \xi $ is an $ n $- dimensional vector $ ( \xi _ {1} \dots \xi _ {n} ) $; and $ \xi ^ {\alpha ^ \prime } = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} } $.

See also Degenerate partial differential equation and the references given there.