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 x x − β u x + 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 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 { σ ϕ } σ > 0 (where ϕ is some compactly supported positive function), we show that: (1) in case β ∈ ( 0 , c 0 ) , there exists σ ⁎ ⩾ 0 such that spreading happens when σ > σ ⁎ (i.e., u ( t , ⋅ ; σ ϕ ) → 1 locally uniformly in R ) and vanishing happens when σ ∈ ( 0 , σ ⁎ ] (i.e., [ g ( t ) , h ( t ) ] remains bounded and u ( t , ⋅ ; σ ϕ ) → 0 uniformly in [ g ( t ) , h ( t ) ] ); (2) in case β ∈ ( c 0 , β ⁎ ) , there exists σ ⁎ > 0 such that virtual spreading happens when σ > σ ⁎ (i.e., u ( t , ⋅ ; σ ϕ ) → 0 locally uniformly in [ g ( t ) , ∞ ) and u ( t , ⋅ + c t ; σ ϕ ) → 1 locally uniformly in R for some c > β − c 0 ), vanishing happens when σ ∈ ( 0 , σ ⁎ ) , and in the transition case σ = σ ⁎ , u ( t , ⋅ + o ( t ) ; σ ϕ ) → V ⁎ ( ⋅ − ( 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 , ⋅ ; σ ϕ ) → 0 uniformly in [ g ( t ) , h ( t ) ] when σ ∈ ( 0 , σ ⁎ ] ; (4) in case β ⩾ β ⁎ , vanishing happens for any solution.