Abstract
This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in \(\mathbb {R}^{n+2}\) with \(n\ge 1\) under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in \(\mathbb {R}^{n+1}\) with \(n\ge 1\).
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Acknowledgements
The authors are grateful to anonymous referees for their very valuable comments and suggestions helping to the improvement of the original manuscript. The second author was supported by NNSF of China (11371179, 11731005) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2017-27).
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Bu, ZH., Wang, ZC. Multidimensional stability of traveling fronts in combustion and non-KPP monostable equations. Z. Angew. Math. Phys. 69, 12 (2018). https://doi.org/10.1007/s00033-017-0906-5
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DOI: https://doi.org/10.1007/s00033-017-0906-5
Keywords
- Planar traveling front
- V-shaped traveling front
- Combustion nonlinearity
- Non-KPP monostable nonlinearity
- Multidimensional asymptotic stability