  The Vlasov equation is well known to provide a good description of the
dynamics of mean-field systems in the $N \to \infty$ limit. This equation has
an infinity of stationary states and the case of {\it homogeneous} states, for
which the single-particle distribution function is independent of the spatial
variable, is well characterized analytically. On the other hand, the
inhomogeneous case often requires some approximations for an analytical
treatment: the dynamics is then best treated in action-angle variables, and the
potential generating inhomogeneity is generally very complex in these new
variables. We here treat analytically the linear stability of toy-models where
the inhomogeneity is created by an external field. Transforming the Vlasov
equation into action-angle variables, we derive a dispersion relation that we
accomplish to solve for both the growth rate of the instability and the
stability threshold for two specific models: the Hamiltonian Mean-Field model
with additional asymmetry and the mean-field $\phi^4$ model. The results are
compared with numerical simulations of the $N$-body dynamics. When the {\it
inhomogeneous} state is stationary, we expect to observe in the $N$-body
dynamics Quasi-Stationary-States (QSS), whose lifetime diverge algebraically
with $N$.
