Michael Beals

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.

Nonlinear Microlocal Analysis

The prototype of the equations described in the introduction is the simple semilinear wave equation

Conormal Regularity after Nonlinear Interaction

U = \{ \partial_t^2 - \sum\limits_{{i = 1}}^n {\partial_{{{x_j}}}^2} \} U = f\left( {t,X,U} \right).

Conormal Waves on Domains with Boundary

(1.1) with f an arbitrary smooth function. We consider first the linear case

Lp Estimates for the wave equation with a potential

U = 0

Optimal L∞ decay for solutions to the wave equation with a potential

(1.2) and several elementary examples of solutions.