SIAM Journal on Mathematical Analysis, Volume 53, Issue 2, Page 2206-2242, January 2021.
We study nonlinear stability of pulled fronts in scalar parabolic equations on the real line of arbitrary order, under conceptual assumptions on existence and spectral stability of fronts. In this general setting, we establish sharp algebraic decay rates and temporal asymptotics of perturbations to the front. Some of these results are known for the specific example of the Fisher-KPP equation, and our results can thus be viewed as establishing universality of some aspects of this simple model. We also give a precise description of how the spatial localization of perturbations to the front affects the temporal decay rate, across the full range of localizations for which asymptotic stability holds. Technically, our approach is based on a detailed study of the resolvent operator for the linearized problem, through which we obtain sharp linear time decay estimates that allow for a direct nonlinear analysis.

中文翻译：

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通过基本光谱附近的溶剂扩展临界拉动前沿的渐近稳定性

SIAM数学分析杂志，第53卷，第2期，第2206-2242页，2021年1月。
我们在关于前沿的存在和谱稳定性的概念性假设下，研究任意阶实线上标量抛物线方程中拉前沿的非线性稳定性。在这种一般情况下，我们建立了急剧的代数衰减率和对前部扰动的时间渐近性。这些结果中的一些对于Fisher-KPP方程的特定示例是已知的，因此我们的结果可以被视为建立了此简单模型某些方面的通用性。我们还精确描述了扰动在空间上的局限性如何影响渐近稳定性的整个局域范围内的时间衰减率。从技术上讲，我们的方法是基于对线性化问题的分解算子的详细研究，