Entire Solutions of the Fisher-KPP Equation on the Half Line

TL;DR

This paper investigates entire solutions of the Fisher-KPP equation on a half line with boundary conditions, constructing solutions that connect traveling waves or scalar solutions to stationary states over time.

Contribution

It establishes the existence of new entire solutions connecting traveling waves or scalar solutions to stationary solutions on a half line.

Findings

Existence of solutions connecting traveling waves to stationary states for c ≥ 2√f'(0).

Construction of solutions linking scalar solutions to stationary states.

Results extend understanding of Fisher-KPP dynamics on half line.

Abstract

In this paper we study the entire solutions of the Fisher-KPP equation $u_t=u_{xx}+f(u)$ on the half line $[0,\infty)$ with Dirichlet boundary condition at $x=0$. (1). For any $c\geq 2\sqrt{f'(0)}$, we show the existence of an entire solution $\mathcal{U}^c(x,t)$ which connects the traveling wave solution $\phi^c(x+ct)$ at $t=-\infty$ and the unique positive stationary solution $V(x)$ at $t=+\infty$; (2). We also construct an entire solution $\mathcal{U}(x,t)$ which connects the solution of $\eta_t =f(\eta)$ at $t=-\infty$ and $V(x)$ at $t=+\infty$.