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.
Publié le : 2004-09-14
Classification:
Geometric optics,
short pulses,
focusing,
caustic,
high frequency asymptotics,
35L70,
35B25,
35B33,
35B40,
35L20,
35Q60
@article{1113246675,
author = {Carles, R\'emi and Rauch, Jeffrey},
title = {Focusing of spherical nonlinear pulses in {$R\sp {1+3}$}, III. Sub and supercritical cases},
journal = {Tohoku Math. J. (2)},
volume = {56},
number = {1},
year = {2004},
pages = { 393-410},
language = {en},
url = {http://dml.mathdoc.fr/item/1113246675}
}
Carles, Rémi; Rauch, Jeffrey. Focusing of spherical nonlinear pulses in {$R\sp {1+3}$}, III. Sub and supercritical cases. Tohoku Math. J. (2), Tome 56 (2004) no. 1, pp. 393-410. http://gdmltest.u-ga.fr/item/1113246675/