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

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 $\beta^*$ with $\beta^*>c_0>0$ 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 $\{ \sigma \phi \}_{\sigma >0}$ (where $\phi$ is some compactly supported positive function), we show that, (1) in case $\beta\in (0,c_0)$, there exists $\sigma^*\geqslant0$ such that spreading happens when $\sigma > \sigma^*$ and vanishing happens when $\sigma \in (0,\sigma^*]$; (2) in case $\beta\in (c_0,\beta^*)$, there exists $\sigma^*>0$ such that virtual spreading happens when $\sigma>\sigma^*$ (i.e., $u(t,\cdot;\sigma \phi)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;\sigma \phi )\to 1$ locally uniformly in $\R$ for some $c>\beta -c_0$), vanishing happens when $\sigma\in (0,\sigma^*)$, and in the transition case $\sigma=\sigma^*$, $u(t, \cdot+o(t);\sigma \phi)\to V^*(\cdot-(\beta-c_0)t )$ 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 $\sigma^*>0$ such that virtual spreading happens when $\sigma > \sigma^*$ and $u(t,\cdot;\sigma \phi)\to 0$ uniformly in $[g(t),h(t)]$ when $\sigma \in (0,\sigma^*]$; (4) in case $\beta\geqslant \beta^*$, vanishing happens for any solution.