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

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 $t^{-3\mathrm{/}2}$ in a weighted $L^{\mathrm{\infty }}$ space. Our proof is based on pointwise semigroup methods and the key remark that the faster algebraic decay rate $t^{-3\mathrm{/}2}$ 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.