## Tohoku Mathematical Journal 2004 September SECOND SERIES VOL. 56, NO. 3

 Tohoku Math. J. 56 (2004), 393-410

Title FOCUSING OF SPHERICAL NONLINEAR PULSES IN $\boldsymbol{R}^{1+3}$, III. SUB AND SUPERCRITICAL CASES

Author Rémi Carles and Jeffrey Rauch

Abstract. We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, then strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded.