Propagation of singularities
Propagation of singularities concerns the extent to which the singular set (or wave front set) of a solution to a PDE at some time t is determined by the singular set at time zero. In physics, Huygens' principle, which asserts that the wave front of a solution propagates at the speed of the medium, is a model example of a propagation of singularities result.
This theory is extremely well developed for variable-coefficient linear equations. For nonlinear equations with mild nonlinearity (such as semilinear equations), what typically happens is that the nonlinearity either has no effect on the propagation of singularities, or only creates some secondary singular sets of weaker strength. The situation is significantly more complicated for quasilinear equations such as the Einstein equations.