Regularity of Nonlinear Waves Associated with a Cusp

We consider local solutions to second order partial differential equations of the form Pu = f(x, u), for which u is smooth on the complement of a characteristic surface with a cusp singularity. If P is strictly hyperbolic and u is assumed to be regular in the past with respect to differentiation by a natural family of smooth vector fields, then u is regular in the future, and “conormal” with respect to a larger family of vector fields which are nonsmooth at the singularity of the cusp. If P is a Tricomi operator associated with the cusp, and the natural initial data (Dirichlet or Cauchy) are conormal with respect to a hyperplane, then u is again shown to be conormal with respect to the cusp.