  We undertake a rigorous study of the stability of the Cauchy horizon in the
naked self-similar Lemaitre-Tolman-Bondi spacetimes under even parity linear
perturbations. We use a combination of energy methods and results about
L^p-spaces to determine the behaviour of the perturbations as they evolve
through the spacetime. We first establish that an average of the perturbation
generically diverges on the Cauchy horizon. We next introduce a rescaled
version of the perturbation, and show that it is bounded and non-zero on the
Cauchy horizon. This in turn shows that the perturbation itself diverges in a
pointwise fashion on the Cauchy horizon. We give a physical interpretation of
this result using the perturbed Weyl scalars. This result supports the
hypothesis of cosmic censorship.
