Entire Solutions of the Fisher-KPP Equation on the 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 $c\ge 2\sqrt{f^{\prime }(0)}$, we show the existence of an entire solution ${\mathcal{U}}^c(x,t)$ which connects the traveling wave solution ${\varphi }^c(x+ct)$ 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 $t=+\infty$.