  The Hunter-Saxton equation serves as a mathematical model for orientation
waves in a nematic liquid crystal. The present paper discusses a modified
variant of this equation, coming up in the study of critical points for the
speed of orientation waves, as well as a two-component extension. We establish
well-posedness and blow-up results for some initial boundary value problems for
the modified Hunter-Saxton equation and the two-component Hunter-Saxton system.
