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- W3019080257 abstract "We study a free boundary problem of the form: u t = u xx + f ( t , u ) ( g ( t ) < x < h ( t )) with free boundary conditions h ′( t ) = − u x ( t , h ( t )) – α ( t ) and g ′( t ) = − u x ( t , g ( t )) + β ( t ), where β ( t ) and α ( t ) are positive T -periodic functions, f ( t , u ) is a Fisher–KPP type of nonlinearity and T -periodic in t . This problem can be used to describe the spreading of a biological or chemical species in time-periodic environment, where free boundaries represent the spreading fronts of the species. We study the asymptotic behaviour of bounded solutions. There are two T -periodic functions α 0 ( t ) and α *( t ; β ) with 0 < α 0 < α * which play key roles in the dynamics. More precisely, (i) in case 0 < β < α 0 and 0 < α < α *, we obtain a trichotomy result: (i-1) spreading, that is, h ( t ) – g ( t ) → +∞ and u ( t , ⋅ + ct ) → 1 with $cin (-overline{l},overline{r})$ , where $ overline{l}:=frac{1}{T}int_{0}^{T}l(s)ds$ , $overline{r}:=frac{1}{T}int_{0}^{T}r(s)ds$ , the T -periodic functions − l ( t ) and r ( t ) are the asymptotic spreading speeds of g ( t ) and h ( t ) respectively (furthermore, r ( t ) > 0 > − l ( t ) when 0 < β < α < α 0 ; r ( t ) = 0 > − l ( t ) when 0 < β < α = α 0 ; $0 gt overline{r} gt -overline{l}$ when 0 < β < α 0 < α < α *); (i-2) vanishing, that is, $limlimits_{t to mathcal {T}}h(t) = limlimits_{t to mathcal {T}}g(t)$ and $limlimits_{t to mathcal {T}}maxlimits_{g(t)leq xleq h(t)} u(t,x)=0$ , where $mathcal {T}$ is some positive constant; (i-3) transition, that is, g ( t ) → −∞, h ( t ) → −∞, $0<limlimits_{t to infty}[h(t)-g(t)] lt +infty$ and u ( t , ⋅) → V ( t , ⋅), where V is a T -periodic solution with compact support. (ii) in case β ≥ α 0 or α ≥ α *, vanishing happens for any solution." @default.
- W3019080257 created "2020-05-01" @default.
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- W3019080257 date "2019-03-25" @default.
- W3019080257 modified "2023-10-17" @default.
- W3019080257 title "Asymptotic behaviour of solutions of Fisher–KPP equation with free boundaries in time-periodic environment" @default.
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- W3019080257 doi "https://doi.org/10.1017/s095679251900010x" @default.
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