  We study a simple system of two hyperbolic semi-linear equations, inspired by
the Einstein equations. The system, which was introduced in gr-qc/0612136, is a
model for singularity formation inside black holes. We show for a particular
case of the equations that the system demonstrates a finite time blowup. The
singularity that is formed is a null singularity. Then we show that in this
particular case the singularity has features that are analogous to known
features of models of black-hole interiors - which describe the inner-horizon
instability. Our simple system may provide insight into the formation of null
singularities inside spinning or charged black holes.
