  We show that the linearized Vlasov-Poisson equations around self-similar
non-homogeneous states near zero contain the full plasma echo mechanism,
yielding Gevrey 3 as a critical stability class. Moreover, here Landau damping
may persist despite blow-up: We construct a critical Gevrey regularity class in
which the force field converges in $L^2$. Thus, on the one hand, the physical
phenomenon of Landau damping holds. On the other hand, the density diverges to
infinity in Sobolev regularity. Hence, ``strong damping'' cannot hold.
