On Echo Chains in Landau damping: Self-similar Solutions and Gevrey 3 as a Linear Stability Threshold

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.