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

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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.
    Mathematics Subject Classification: Primary: 35B35, 35A18; Secondary: 35K57, 35B40.


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