Explosion et normes $L^p$ pour l'equation des ondes non lineaire cubique
Résumé
We find blow-up solutions of nonlinear wave equations with cubic
nonlinearity, in any number of space dimensions, and study the
asymptotic behavior of their $L^p$ norms and ``energy.'' The $L^p$
norm blows up if the blow-up surface has an interior
non-degenerate minimum and $p\geq n/2$. For less smooth right-hand
sides, and $0<\varepsilon<1$, we give examples for which the $L^p$ norm
blows up if $p\geq n/(1+\varepsilon)$; their Cauchy data are unbounded,
but blow-up is not instantaneous. Applications to nonlinear optics
are briefly outlined.