Asymptotic Behavior and Error Bounds for Fisher-KPP Equations on the Real Half-Line

Abstract

We study the Fisher--KPP equation on the half-line under Dirichlet,Neumann, and Robin boundary conditions. For the autonomous logistic equation, we identify bounded stationary profiles converging to $1$ and obtain exponential far-field comparison estimates. We prove local uniform convergence of nontrivial Neumann solutions to $1$. Assuming local uniform convergence of the Robin solution to its stationary profile, we derive asymptotic Neumann--Robin comparison estimates. We then consider small time-periodic Neumann and Robin boundary forcing. Under exponential stability of the homogeneous linearized semigroup, we construct a locally unique small periodic lifted mild solution and obtain a first-order expansion with a uniform $O({\epsilon }^2)$ remainder in $C_0([0,\infty ))$.