Survival of the black hole's Cauchy horizon under noncompact perturbations

We study numerically the evolution of spacetime, and in particular of a spacetime singularity, inside a black hole under a class of perturbations of noncompact support. We use a very simplified toy model of a spherical charged black hole which is perturbed nonlinearly by a self-gravitating, spherical scalar field. The latter grows logarithmically with advanced time along an outgoing characteristic hypersurface. We find that for that class of perturbations a portion of the Cauchy horizon survives as a noncentral, null singularity.