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Asymptotic behaviour of solutions of Fisher–KPP equation with free boundaries in time-periodic environment

Part of: Parabolic equations and systems Partial differential equations Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  25 March 2019

JINGJING CAI  and
LI XU
JINGJING CAI
Affiliation:
School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai200090, China. emails: cjjing1983@163.com; xulimaths@163.com
LI XU
Affiliation:
School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai200090, China. emails: cjjing1983@163.com; xulimaths@163.com
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Abstract

We study a free boundary problem of the form: ut = uxx + f(t, u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) – α(t) and g′(t) = −ux(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 α0(t) and α*(t; β) with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β< α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, that is, h(t) – g(t) → +∞ and u(t, ⋅ + ct) → 1 with $c\in (-\overline{l},\overline{r})$, 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 < α0; r(t) = 0 > −l(t) when 0 < β < α = α0; $0 \gt \overline{r} \gt -\overline{l}$ when 0 < β < α0 < α < α*); (i-2) vanishing, that is, $\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$ and $\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$, where $\mathcal {T}$ is some positive constant; (i-3) transition, that is, g(t) → −∞, h(t) → −∞, $0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$ and u(t, ⋅) → V(t, ⋅), where V is a T-periodic solution with compact support. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

Keywords

Fisher–KPP equationfree boundary problemperiodic travelling waveasymptotic behaviour

MSC classification

Primary: 35K20: Initial-boundary value problems for second-order parabolic equations
Secondary: 35K55: Nonlinear parabolic equations 35B40: Asymptotic behavior of solutions 35R35: Free boundary problems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 3 , June 2020 , pp. 423 - 449
Copyright
© Cambridge University Press 2019

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Footnotes

†

This research was sponsored by NSFC (No. 11701359, No. 11502141).

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