Asymptotic stability of the critical Fisher-KPP front using pointwise estimates

Grégory Faye (1) , Matt Holzer (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 George Mason University [Fairfax]

We propose a simple alternative proof of a famous result of Gallay regarding the nonlinear asymptotic stability of the critical front of the Fisher-KPP equation which shows that perturbations of the critical front decay algebraically with rate $t^{-3/2}$ in a weighted $L^\infty$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3/2}$ is a consequence of the lack of an embedded zero of the Evans function at the origin for the linearized problem around the critical front.

Mots clés: pointwise Green's function; Fisher-KPP equation; nonlinear stability

Domaines: Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Grégory Faye, Matt Holzer. Asymptotic stability of the critical Fisher-KPP front using pointwise estimates. Zeitschrift für Angewandte Mathematik und Physik, In press. ⟨hal-01676017⟩