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Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions


Authors: Frank Merle and Hatem Zaag
Journal: Trans. Amer. Math. Soc. 368 (2016), 27-87
MSC (2010): Primary 35L05, 35L71, 35L67, 35B44, 35B40
DOI: https://doi.org/10.1090/tran/6450
Published electronically: April 15, 2015
MathSciNet review: 3413856
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Abstract: This is the first of two papers devoted to the study of the properties of the blow-up surface for the $ N$ dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the space-independent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global or go to zero or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution.


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Additional Information

Frank Merle
Affiliation: Département de Mathématiques, Université de Cergy Pontoise and IHES, 2 avenue Adolphe Chauvin, BP 222, F-95302 Cergy Pontoise cedex, France
Email: merle@math.u-cergy.fr

Hatem Zaag
Affiliation: Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), 99 avenue J.B. Clément, F-93430 Villetaneuse, France
Email: Hatem.Zaag@univ-paris13.fr

DOI: https://doi.org/10.1090/tran/6450
Keywords: Semilinear wave equation, blow-up, higher-dimensional case
Received by editor(s): October 7, 2013
Published electronically: April 15, 2015
Additional Notes: Both authors were supported by the ERC Advanced Grant no. 291214, BLOWDISOL
The second author was partially supported by the ANR Project ANAÉ ref. ANR-13-BS01-0010-03.
Article copyright: © Copyright 2015 American Mathematical Society