In this paper, for degenerate n-degree Fisher-type equations, we discuss the stability of their traveling front solutions with noncritical speeds. In fact, when the initial perturbations around these noncritical traveling front solutions are in some weighted Banach spaces, we have proved that these solutions are globally exponentially stable in the form of ${(1+t)}^{\frac{1}{3}}{e}^{-\nu t}$ for ν ∈ (0, 1) via L¹-energy estimates, L²-energy estimates, and the weighted energy method. Furthermore, by Fourier transform and the weighted energy method, we will prove that traveling front solutions with noncritical speeds are also globally exponentially stable in the form of $t^{-\frac{1}{2}}{e}^{-\nu t}$ for some positive constant ν when the initial perturbations around these solutions are in some weighted Sobolev spaces. Our conclusions extend the local stability of noncritical traveling front solutions into the global case and also give some novel forms of exponential stability of these solutions.

中文翻译：

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具有退化非线性的Fisher型方程非临界行进前沿解的全局稳定性

在本文中，对于退化的n度 Fisher 型方程，我们讨论了它们在非临界速度下行进前沿解的稳定性。事实上，当围绕这些非临界行进前沿解的初始扰动处于某些加权 Banach 空间时，我们已经证明这些解在以下形式中是全局指数稳定的${(1+吨)}^{\frac{1}{3}}{\mathrm{电子}}^{-\nu 吨}$对于ν ∈ (0, 1) 通过L ^{1 -}能量估计、L ^{2 -}能量估计和加权能量方法。此外，通过傅里叶变换和加权能量方法，我们将证明具有非临界速度的行进前沿解也是全局指数稳定的，形式为${吨}^{-\frac{1}{2}}{\mathrm{电子}}^{-\nu 吨}$当围绕这些解的初始扰动在某些加权 Sobolev 空间中时，对于某些正常数ν。我们的结论将非关键旅行前沿解决方案的局部稳定性扩展到整体情况下，并且给出了这些解决方案的指数稳定性的一些新颖形式。