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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions
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by Frank Merle and Hatem Zaag PDF
Trans. Amer. Math. Soc. 368 (2016), 27-87 Request permission

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
  • MR Author ID: 123710
  • 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
  • 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.
  • © Copyright 2015 American Mathematical Society
  • 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
  • MathSciNet review: 3413856