Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension

Merle, Frank ¹     ; Zaag, Hatem ²  

¹ Université de Cergy Pontoise Département de mathématiques 2 avenue Adolphe Chauvin BP 222 95302 Cergy Pontoise cedex France
² Université Paris 13, Institut Galilée Laboratoire Analyse, Géométrie et Applications CNRS UMR 7539 99 avenue J.B. Clément 93430 Villetaneuse France

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2009-2010), Exposé no. 11, 10 p.

Résumé

We consider the semilinear wave equation with power nonlinearity in one space dimension. We first show the existence of a blow-up solution with a characteristic point. Then, we consider an arbitrary blow-up solution $u(x,t)$, the graph $x\mapsto T\left(x\right)$ of its blow-up points and $𝒮\subset ℝ$ the set of all characteristic points and show that $𝒮$ is locally finite. Finally, given $x_0\in 𝒮$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that $T\left(x\right)$ forms a corner of angle $\frac{\pi }{2}$.