High regularity waves on self-similar naked singularity interiors: decay and the role of blue-shift

We consider solutions to the linear wave equation $\Box_{g}\varphi = 0$ on a class of approximately $k$-self-similar naked singularity interiors. This equation models the blue-shift effect, an instability exploited by Christodoulou in the proof of low-regularity weak cosmic censorship. Using a combination of resonance expansions and multiplier estimates, we find in the small-mass regime $k^2 \ll 1$ that the asymptotics of solutions are strongly sensitive to the regularity assumed on outgoing, characteristic initial data across the past light-cone of the singularity. Above a threshold regularity set by the $k$-self-similar scalar field, solutions are shown to always obey self-similar bounds, indicating that the blue-shift instability competes with the stabilizing influence of high regularity. We conclude that a proper statement of weak cosmic censorship, as well as an understanding of the role of naked singularities in phenomena such as critical collapse, may depend on the topology of initial data.