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

Abstract

This paper is concerned with the asymptotic stability oftravelling 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 isproved to be locally exponentially stable to perturbations in someexponentially weighted $L^{\infty}$ spaces. Further by Evansfunction method and detailed semigroup estimates, the travellingwave fronts with non-critical speed are proved to be locallyalgebraically stable to perturbations in some polynomiallyweighted $L^{\infty}$ spaces. It's remarked that due to the slowalgebraic decay rate of the wave at $+\infty,$ the Evans functionconstructed in this paper is an extension of the definitions in[1, 3, 7, 11, 21] to some extent, and the Evansfunction can be extended analytically in the neighborhood of theorigin.