Communications in Mathematical Physics

, 282:55

Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation


  • Frank Merle
    • Département de mathématiquesUniversité de Cergy Pontoise, IHES and CNRS
    • Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539

DOI: 10.1007/s00220-008-0532-3

Cite this article as:
Merle, F. & Zaag, H. Commun. Math. Phys. (2008) 282: 55. doi:10.1007/s00220-008-0532-3


We consider here the 1 D semilinear wave equation with a power nonlinearity and with no restriction on initial data. We first prove a Liouville Theorem for that equation. Then, we consider a blow-up solution, its blow-up curve \({x\mapsto T(x)}\) and \({I_0\subset \mathbb{R}}\) the set of non-characteristic points. We show that I0 is open and that T(x) is C1 on I0. All these results fundamentally use our previous result in [19] showing the convergence in selfsimilar variables for \({x\in I_0}\) .

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© Springer-Verlag 2008