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

We consider Fisher-KPP equation with advection: $u_t=u_{xx}-βu_x+f(u)$ 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 $-β$ on the long time behavior of the solutions. We find two parameters $c_0$ and $β^*$ with $β^*>c_0>0$ 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 $\{ σϕ\}_{σ>0}$ (where $ϕ$ is some compactly supported positive function), we show that, (1) in case $β\in (0,c_0)$, there exists $σ^*\geqslant0$ such that spreading happens when $σ> σ^*$ and vanishing happens when $σ\in (0,σ^*]$; (2) in case $β\in (c_0,β^*)$, there exists $σ^*>0$ such that virtual spreading happens when $σ>σ^*$ (i.e., $u(t,\cdot;σϕ)\to 0$ locally uniformly in $[g(t),\infty)$ and $u(t,\cdot + ct;σϕ)\to 1$ locally uniformly in $\R$ for some $c>β-c_0$), vanishing happens when $σ\in (0,σ^*)$, and in the transition case $σ=σ^*$, $u(t, \cdot+o(t);σϕ)\to V^*(\cdot-(β-c_0)t )$ uniformly, the latter is a traveling wave with a "big head" near the free boundary $x=(β-c_0)t$ and with an infinite long "tail" on the left; (3) in case $β= c_0$, there exists $σ^*>0$ such that virtual spreading happens when $σ> σ^*$ and $u(t,\cdot;σϕ)\to 0$ uniformly in $[g(t),h(t)]$ when $σ\in (0,σ^*]$; (4) in case $β\geqslant β^*$, vanishing happens for any solution.