  Einstein's equations are known to lead to the formation of black holes and
spacetime singularities. This appears to be a manifestation of the mathematical
phenomenon of finite-time blowup: a formation of singularities from regular
initial data. We present a simple hyperbolic system of two semi-linear
equations inspired by the Einstein equations. We explore a class of solutions
to this system which are analogous to static black-hole models. These solutions
exhibit a black-hole structure with a finite-time blowup on a characteristic
line mimicking the null inner horizon of spinning or charged black holes. We
conjecture that this behavior - namely black-hole formation with blow-up on a
characteristic line - is a generic feature of our semi-linear system. Our
simple system may provide insight into the formation of null singularities
inside spinning or charged black holes in the full system of Einstein
equations.
