Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries

Abstract

We consider Fisher-KPP equation with advection: $u_t=u_{xx}-\beta u_x+f(u)$ for $x\in (g(t),h(t))$, where $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation is used to describe the population dynamics in advective environments. We study the influence of the advection coefficient $-\beta$ on the long time behavior of the solutions. We find two parameters $c_0$ and ${\beta }^{\ast }$ with ${\beta }^{\ast }>c_0>0$ which play key roles in the dynamics, here $c_0$ is the minimal speed of the traveling waves of Fisher-KPP equation. More precisely, by studying a family of the initial data $\{\sigma \varphi {\}}_{\sigma >0}$ (where $\varphi$ is some compactly supported positive function), we show that, (1) in case $\beta \in (0,c_0)$, there exists ${\sigma }^{\ast }⩾0$ such that spreading happens when $\sigma >{\sigma }^{\ast }$ and vanishing happens when $\sigma \in (0,{\sigma }^{\ast }]$; (2) in case $\beta \in (c_0,{\beta }^{\ast })$, there exists ${\sigma }^{\ast }>0$ such that virtual spreading happens when $\sigma >{\sigma }^{\ast }$ (i.e., $u(t,\cdot ;\sigma \varphi )\to 0$ locally uniformly in $[g(t),\infty )$ and $u(t,\cdot +ct;\sigma \varphi )\to 1$ locally uniformly in $\mathbb{R}$ for some $c>\beta -c_0$), vanishing happens when $\sigma \in (0,{\sigma }^{\ast })$, and in the transition case $\sigma ={\sigma }^{\ast }$, $u(t,\cdot +o(t);\sigma \varphi )\to V^{\ast }(\cdot -(\beta -c_0)t)$ uniformly, the latter is a traveling wave with a "big head" near the free boundary $x=(\beta -c_0)t$ and with an infinite long "tail" on the left; (3) in case $\beta =c_0$, there exists ${\sigma }^{\ast }>0$ such that virtual spreading happens when $\sigma >{\sigma }^{\ast }$ and $u(t,\cdot ;\sigma \varphi )\to 0$ uniformly in $[g(t),h(t)]$ when $\sigma \in (0,{\sigma }^{\ast }]$; (4) in case $\beta ⩾{\beta }^{\ast }$, vanishing happens for any solution.

41 pages, 6 figures

References in corpus (1)

• Different Asymptotic Spreading Speeds Induced by Advection in a Diffusion Problem with Free Boundaries