  We prove that a large class of smooth solutions $\psi$ to the linear wave
equation $\Box_g\psi=0$ on subextremal rotating Kerr spacetimes which are
regular and decaying along the event horizon become singular at the Cauchy
horizon. More precisely, we show that assuming appropriate upper and lower
bounds on the energy along the event horizon, the solution has infinite
(non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy
horizon transversally. Extrapolating from known results in the
Reissner--Nordstr\"om case, the assumed upper and lower bounds required for our
theorem are conjectured to hold for solutions arising from generic smooth and
compactly supported initial data on a Cauchy hypersurface. This result is
motivated by the strong cosmic censorship conjecture in general relativity.
