# 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 $\xi ^ {\alpha ^ \prime } = \xi _ {1} ^ {\alpha _ {1} } \dots \xi _ {n} ^ {\alpha _ {n} }$.