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Almost sharp nonlinear scattering in one-dimensional Born-Infeld equations arising in nonlinear electrodynamics

Authors: Miguel A. Alejo and Claudio Muñoz
Journal: Proc. Amer. Math. Soc. 146 (2018), 2225-2237
MSC (2010): Primary 37K15, 35Q53; Secondary 35Q51, 37K10
Published electronically: January 8, 2018
MathSciNet review: 3767373
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Abstract: We study decay of small solutions of the Born-Infeld equation in 1+1 dimensions, a quasilinear scalar field equation modeling nonlinear electromagnetism, as well as branes in String theory and minimal surfaces in Minkowski space-times. From the work of Whitham, it is well known that there is no decay because of arbitrary solutions traveling to the speed of light just as linear wave equation. However, even if there is no global decay in 1+1 dimensions, we are able to show that all globally small $H^{s+1}\times H^s$, $s>\frac 12$ solutions do decay to the zero background state in space, inside a strictly proper subset of the light cone. We prove this result by constructing a Virial identity related to a momentum law, in the spirit of works by Kowalczyk, Martel, and the second author, as well as a Lyapunov functional that controls the $\dot H^1 \times L^2$ energy.

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Additional Information

Miguel A. Alejo
Affiliation: Departamento de Matemática, Universidade Federal de Santa Catarina, Brasil
MR Author ID: 975414

Claudio Muñoz
Affiliation: Departamento de Ingeniería Matemática and CMM UMI 2807-CNRS , Universidad de Chile, Santiago, Chile
MR Author ID: 806855

Keywords: Born-Infeld equation, scattering, decay estimates, Virial
Received by editor(s): July 9, 2017
Received by editor(s) in revised form: August 22, 2017
Published electronically: January 8, 2018
Additional Notes: The first author was partially funded by Product. CNPq grant no. 305205/2016-1, Universal 16 CNPq grant no. 431231/2016-8 and MathAmSud/Capes EEQUADD collaboration Math16-01.
The second author was partially supported by Fondecyt no. 1150202, Millennium Nucleus Center for Analysis of PDE NC130017, Fondo Basal CMM, and MathAmSud EEQUADD collaboration Math16-01.
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society