  This paper contributes to the wider study of hyperbolic equations with
multiplicities. We focus here on some classes of higher order hyperbolic
equations with space dependent coefficients in any space dimension. We prove
Sobolev well-posedness of the corresponding Cauchy problem (with loss of
derivatives due to the multiplicities) under suitable Levi conditions on the
lower order terms. These conditions generalise the well known Olienik's
conditions in \cite{O70} to orders higher than $2$.
