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Blow-up profile for the complex-valued semilinear wave equation


Author: Asma Azaiez
Journal: Trans. Amer. Math. Soc. 367 (2015), 5891-5933
MSC (2010): Primary 35L05, 35L81, 35B44, 39B32, 35B40, 34K21, 35B35
DOI: https://doi.org/10.1090/S0002-9947-2014-06370-8
Published electronically: August 12, 2014
MathSciNet review: 3347192
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Abstract: In this paper, we consider a blow-up solution for the complex-valued semilinear wave equation with power non-linearity in one space dimension. We first characterize all the solutions of the associated stationary problem as a two-parameter family. Then, we use a dynamical system formulation to show that the solution in self-similar variables approaches some particular stationary one in the energy norm, in the non-characteristic case. This gives the blow-up profile for the original equation in the non-characteristic case. Our analysis is not just a simple adaptation of the already handled real case. In particular, there is one more neutral-direction in our problem, which we control thanks to a modulation technique.


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Additional Information

Asma Azaiez
Affiliation: Institut Galilée, Laboratoire Analyse Géometrie et Applications, CNRS-UMR 7539, Université Paris 13, 99 avenue J.B. Clément 93430, Villetaneuse, France
Email: azaiez@math.univ-paris13.fr

DOI: https://doi.org/10.1090/S0002-9947-2014-06370-8
Keywords: Wave equation, blow-up profile, stationary solution, modulation technique, complex-valued PDE
Received by editor(s): June 19, 2013
Received by editor(s) in revised form: October 25, 2013, and November 30, 2013
Published electronically: August 12, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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