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We study a simple system of two hyperbolic semi-linear equations, inspired bythe Einstein equations. The system, which was introduced in gr-qc/0612136, is amodel for singularity formation inside black holes. We show for a particularcase of the equations that the system demonstrates a finite time blowup. Thesingularity that is formed is a null singularity. Then we show that in thisparticular case the singularity has features that are analogous to knownfeatures of models of black-hole interiors - which describe the inner-horizoninstability. Our simple system may provide insight into the formation of nullsingularities inside spinning or charged black holes.Comment: 25 pages, 10 figure

Simplified mathematical model for the formation of null singularities inside black holes. II. Proof for a particular case