Transition Fronts of Combustion Reaction Diffusion Equations in \(\mathbb {R}^{N}\)

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Abstract

This paper is concerned with combustion transition fronts in \(\mathbb {R}^{N}\) \((N\ge 1)\). Firstly, we prove the existence and the uniqueness of the global mean speed which is independent of the shape of the level sets of the fronts. Secondly, we show that the planar fronts can be characterized in the more general class of almost-planar fronts. Thirdly, we show the existence of new types of transitions fronts in \(\mathbb {R}^{N}\) which are not standard traveling fronts. Finally, we prove that all transition fronts are monotone increasing in time, whatever shape their level sets may have.

Keywords

Reaction–diffusion equations Combustion nonlinearity Transition front Qualitative properties 

Mathematics Subject Classification

35K40 35K57 35C07 35B35 35B40 

Notes

Acknowledgements

The first author and the second author would like to thank Professor François Hamel of Aix-Marseille University for the valuable discussions. They was supported by the China Scholarship Council. The third author was supported by NNSF of China (11371179, 11731005) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2016-ct12, lzujbky-2017-ct01). This work has been partly carried out in the framework of the ANR DEFI Project NONLOCAL (ANR-14-CE25-0013), of Archimèdes Labex (ANR-11-LABX-0033), of the A*MIDEX Project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR), and of the ERC Project ReaDi - Reaction–Diffusion Equations, Propagation and Modelling, Grant Agreement n. 321186 funded by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.CNRS, Centrale Marseille, I2MAix Marseille UnivMarseilleFrance

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