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

We consider Fisher-KPP equation with advection: ut=uxx−βux+f(u)ut=uxx−βux+f(u) for x∈(g(t),h(t))x∈(g(t),h(t)), where g(t)g(t) and h(t)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 c0c0 and β⁎β⁎ with β⁎>c0>0β⁎>c0>0 which play key roles in the dynamics, here c0c0 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{σϕ}σ>0 (where ϕ   is some compactly supported positive function), we show that: (1) in case β∈(0,c0)β∈(0,c0), there exists σ⁎⩾0σ⁎⩾0 such that spreading happens when σ>σ⁎σ>σ⁎ (i.e., u(t,⋅;σϕ)→1 locally uniformly in RR) and vanishing happens when σ∈(0,σ⁎]σ∈(0,σ⁎] (i.e., [g(t),h(t)][g(t),h(t)] remains bounded and u(t,⋅;σϕ)→0 uniformly in [g(t),h(t)][g(t),h(t)]); (2) in case β∈(c0,β⁎)β∈(c0,β⁎), there exists σ⁎>0σ⁎>0 such that virtual spreading happens when σ>σ⁎σ>σ⁎ (i.e., u(t,⋅;σϕ)→0 locally uniformly in [g(t),∞)[g(t),∞) and u(t,⋅+ct;σϕ)→1u(t,⋅+ct;σϕ)→1 locally uniformly in RR for some c>β−c0c>β−c0), vanishing happens when σ∈(0,σ⁎)σ∈(0,σ⁎), and in the transition case σ=σ⁎σ=σ⁎, u(t,⋅+o(t);σϕ)→V⁎(⋅−(β−c0)t)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)tx=(β−c0)t and with an infinite long “tail” on the left; (3) in case β=c0β=c0, there exists σ⁎>0σ⁎>0 such that virtual spreading happens when σ>σ⁎σ>σ⁎ and u(t,⋅;σϕ)→0 uniformly in [g(t),h(t)][g(t),h(t)] when σ∈(0,σ⁎]σ∈(0,σ⁎]; (4) in case β⩾β⁎β⩾β⁎, vanishing happens for any solution.