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

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Abstract

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 I 0 is open and that T(x) is C 1 on I 0. 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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Authors and Affiliations

  1. Département de mathématiques, Université de Cergy Pontoise, IHES and CNRS, 2 avenue Adolphe Chauvin, BP 222, 95302, Cergy Pontoise cedex, France

    Frank Merle

  2. Université Paris 13, Institut Galilée, Laboratoire Analyse, Géométrie et Applications, CNRS UMR 7539, 99 avenue J.B. Clément, 93430, Villetaneuse, France

    Hatem Zaag

Authors
  1. Frank Merle
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  2. Hatem Zaag
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Correspondence to Hatem Zaag.

Additional information

Communicated by P. Constantin

This work was supported by a grant from the french Agence Nationale de la Recherche, project ONDENONLIN, reference ANR-06-BLAN-0185.

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Merle, F., Zaag, H. Openness of the Set of Non-characteristic Points and Regularity of the Blow-up Curve for the 1 D Semilinear Wave Equation. Commun. Math. Phys. 282, 55–86 (2008). https://doi.org/10.1007/s00220-008-0532-3

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