Spatial Decay and Asymptotic Stability of Forced Waves at the Critical Speed for the Fisher-KPP Equation in a Shifting Environment

In this paper, we investigate the spatial decay and asymptotic stability of forced waves for the Fisher-KPP equation in a shifting heterogeneous environment, focusing on the critical speed $c_{\ast }=2\sqrt{r_{+}}$. Within the framework of regular variation, we establish a complete classification of critical wave tails in terms of the asymptotic behavior of the environmental deviation $r_{+}-r(z)$ as $z\to +\mathrm{\infty }$. In each case, we derive precise spatial decay that characterize how the environmental tail modifies the leading-order decay of the wave at criticality. We further establish asymptotic stability estimates of forced waves. For initial data satisfying an exponentially weighted boundedness condition, we prove an $O(t^{-1\mathrm{/}2})$ convergence rate in the uniform norm. Under stronger spatial decay assumptions on the weighted initial data, this rate improves to $O(t^{-1-\sigma \mathrm{/}2})$, for $\sigma \in [0,1]$, in both the uniform norm and a weighted norm. Moreover, no smallness assumption on the initial perturbations is needed for our results.