Stability of travelling waves with algebraic decay for n-degree Fisher-type equations

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March 2006, 16(1): 47-66. doi: 10.3934/dcds.2006.16.47

Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations

Yaping Wu ^1, , Xiuxia Xing ^2, and Qixiao Ye ^3,

1.	Department of Mathematics, Capital Normal University, Beijing 100037, China
2.	College of Applied Science, Beijing University of Technology, Beijing 100022, China
3.	Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

Received September 2005 Revised February 2006 Published June 2006

Abstract
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This paper is concerned with the asymptotic stability of travelling wave front solutions with algebraic decay for $n$-degree Fisher-type equations. By detailed spectral analysis, each travelling wave front solution with non-critical speed is proved to be locally exponentially stable to perturbations in some exponentially weighted $L^{\infty}$ spaces. Further by Evans function method and detailed semigroup estimates, the travelling wave fronts with non-critical speed are proved to be locally algebraically stable to perturbations in some polynomially weighted $L^{\infty}$ spaces. It's remarked that due to the slow algebraic decay rate of the wave at $+\infty,$ the Evans function constructed in this paper is an extension of the definitions in [1, 3, 7, 11, 21] to some extent, and the Evans function can be extended analytically in the neighborhood of the origin.

Keywords: travelling wave fronts, Asymptotic stability, algebraic decay rate, spectral analysis, semigroup estimates..

Mathematics Subject Classification: Primary: 35B35, 35A18; Secondary: 35K57, 35B4.

Citation: Yaping Wu, Xiuxia Xing, Qixiao Ye. Stability of travelling waves with algebraic decay for $n$-degree Fisher-type equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 47-66. doi: 10.3934/dcds.2006.16.47

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