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

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We consider Fisher-KPP equation with advection: uₜ=uxx-βuₓ+f(u) for x∈ (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 c₀ and β* with β*>c₀>0 which play key roles in the dynamics, here c₀ 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∈ (0,c₀), there exists σ*0 such that spreading happens when σ> σ* and vanishing happens when σ∈ (0,σ*]; (2) in case β∈ (c₀,β*), there exists σ*>0 such that virtual spreading happens when σ>σ* (i.e., u(t,·;σϕ)→ 0 locally uniformly in [g(t),∞) and u(t,· + ct;σϕ)→ 1 locally uniformly in for some c>β-c₀), vanishing happens when σ∈ (0,σ*), and in the transition case σ=σ*, u(t, ·+o(t);σϕ)→ V*(·-(β-c₀)t) uniformly, the latter is a traveling wave with a "big head" near the free boundary x=(β-c₀)t and with an infinite long "tail" on the left; (3) in case β= c₀, there exists σ*>0 such that virtual spreading happens when σ> σ* and u(t,·;σϕ)→ 0 uniformly in [g(t),h(t)] when σ∈ (0,σ*]; (4) in case β β*, vanishing happens for any solution.