Nonlinear hyperbolic smoothing at a focal point.

Abstract

This thesis studies the propagation of singularities of solutions to the semi-linear dissipative wave equation &squ;<italic>u</italic> + |<italic> u_t</italic>|<super>p-1</super><italic>u_t</italic> = 0 with <italic>p</italic> > 1. The main result is the proof that there is a smoothing phenomenon at a spherical focal point for general piecewise smooth Cauchy data when the spatial dimension is sufficiently large. More precisely, the nonlinear wave equation under study has strong solutions which have a focusing wave of singularity on an incoming light cone. The singularity is focused and partially smoothed out at the tip of the light cone. After the focusing time, the solutions are smoother than they were in the sense of Sobolev regularity. This smoothing phenomenon was first studied by J.-L. Joly, G. Metivier and J. Rauch for spherically symmetric Cauchy data. In this thesis, I show that the same smoothing phenomenon occurs for general Cauchy data with the spherical symmetry assumption removed. The core of the proof are the construction of piecewise <italic>H</italic><super>2</super> regularity before the focusing time, justification of a nonlinear transport equation and conical energy estimates. We also find that the asymptotic behavior of (&part;<italic>_t</italic> + &part;<italic>_r</italic>)<italic>u</italic> along the focusing cone is to leading order independent of initial data and can be described by a self-similar radial solution. We prove the existence and uniqueness of such self-similar solutions by analyzing a singular ODE problem. Moreover, the results of the smoothing theorem are sharp at the focus for the self-similar solutions.