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

Frank Merle[1]; Hatem Zaag[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 (2009-2010)

  • Volume: 1996-1997, page 1-10

Abstract

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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 ↦ T ( x ) of its blow-up points and 𝒮 ⊂ ℝ the set of all characteristic points and show that 𝒮 is locally finite. Finally, given x 0 ∈ 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 ( x ) forms a corner of angle π 2 .

How to cite

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Merle, Frank, and Zaag, Hatem. "Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension." Séminaire Équations aux dérivées partielles 1996-1997 (2009-2010): 1-10. <http://eudml.org/doc/116437>.

@article{Merle2009-2010,
abstract = {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 $x\mapsto T(x)$ of its blow-up points and $\mathcal\{S\}\subset \mathbb\{R\} $ the set of all characteristic points and show that $\mathcal\{S\}$ is locally finite. Finally, given $x_0\in \mathcal\{S\}$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that $T(x)$ forms a corner of angle $\frac\{\pi \}\{2\}$.},
affiliation = {Université de Cergy Pontoise Département de mathématiques 2 avenue Adolphe Chauvin BP 222 95302 Cergy Pontoise cedex France; Université Paris 13, Institut Galilée Laboratoire Analyse, Géométrie et Applications CNRS UMR 7539 99 avenue J.B. Clément 93430 Villetaneuse France},
author = {Merle, Frank, Zaag, Hatem},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
pages = {1-10},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension},
url = {http://eudml.org/doc/116437},
volume = {1996-1997},
year = {2009-2010},
}

TY - JOUR
AU - Merle, Frank
AU - Zaag, Hatem
TI - Isolatedness of characteristic points at blow-up for a semilinear wave equation in one space dimension
JO - Séminaire Équations aux dérivées partielles
PY - 2009-2010
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1996-1997
SP - 1
EP - 10
AB - 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 $x\mapsto T(x)$ of its blow-up points and $\mathcal{S}\subset \mathbb{R} $ the set of all characteristic points and show that $\mathcal{S}$ is locally finite. Finally, given $x_0\in \mathcal{S}$, we show that in selfsimilar variables, the solution decomposes into a decoupled sum of (at least two) solitons, with alternate signs and that $T(x)$ forms a corner of angle $\frac{\pi }{2}$.
LA - eng
UR - http://eudml.org/doc/116437
ER -

References

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  18. F. Merle and H. Zaag. Openness of the set of non characteristic points and regularity of the blow-up curve for the 1 d semilinear wave equation. Comm. Math. Phys., 282:55–86, 2008. Zbl1159.35046MR2415473
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  20. F. Merle and H. Zaag. Isolatedness of characteristic points for a semilinear wave equation in one space dimension. 2010. preprint. Zbl1270.35320
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  22. H. Zaag. Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation. Duke Math. J., 133(3):499–525, 2006. Zbl1096.35062MR2228461

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