We consider hyperbolic Cauchy problems with characteristics
of variable multiplicity and coefficients of polynomial growth
in the space variables; we focus on second order equations
and admit finite order intersections between the characteristics.
We obtain well posedness results in $\mathcal{S}(\mathbb{R}^{n})$,
$\mathcal{S}'(\mathbb{R}^{n})$ by imposing suitable Levi conditions
on the lower order terms. By an energy estimate in weighted
Sobolev spaces we show that regularity and behavior at infinity
of the solution are different from the ones of the data.