  We study the zero dispersion limit of the fifth order KdV equations when time
is sufficiently large. In general, the weak limit may be described by an
arbitrary odd number of hyperbolic equations. Unlike the KdV case, these are
non-strictly hyperbolic equations. However, we show that the weak limit is
governed by three hyperbolic equations in a domain in the space-time for all
times bigger than a large time. Outside this domain, the weak limit satisfies a
single hyperbolic equation.
