Explosion et normes [formula omitted] pour l'équation des ondes nonlinéaire cubique

Abstract

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⩾ n/2. For less smooth right-hand sides, and 0< ε<1, we give examples for which the L p norm blows up if p⩾ n/(1+ ε); their Cauchy data are unbounded, but blow-up is not instantaneous. Applications to nonlinear optics are briefly outlined. To cite this article: G. Cabart, S. Kichenassamy, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 903–908.