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.

中文翻译：

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时间周期环境下自由边界Fisher-KPP方程解的渐近行为

我们研究以下形式的自由边界问题：你吨=你xx+F(吨,你) (G(吨) <X<H(吨)) 具有自由边界条件H'(吨) = -你X(吨,H(吨))——α(吨） 和G'(吨) = -你X(吨,G(吨)) +β(吨）， 在哪里β(吨） 和α(吨) 为正吨- 周期函数，F(吨,你) 是 Fisher-KPP 类型的非线性，并且吨- 定期在吨. 这个问题可以用来描述生物或化学物种在时间周期环境中的传播，其中自由边界代表物种的传播前沿。我们研究有界解的渐近行为。那里有两个吨-周期性函数α0(吨） 和α*(吨;β) 与 0 <α0<α* 在动态中起关键作用。更准确地说，(i) 如果 0 <β<α0和 0 <α<α*，我们得到一个三分法结果：（i-1）传播，即H(吨) –G(吨) → +∞ 和你(吨, ⋅ +ct) → 1 与$c\in (-\overline{l},\overline{r})$， 在哪里$ \overline{l}:=\frac{1}{T}\int_{0}^{T}l(s)ds$,$\overline{r}:=\frac{1}{T}\int_{0}^{T}r(s)ds$， 这吨-周期性函数 -l(吨） 和r(吨) 是渐近传播速度G(吨） 和H(吨）分别（此外，r(吨) > 0 > -l(吨) 当 0 <β<α<α0;r(吨) = 0 > -l(吨) 当 0 <β<α=α0;$0 \gt \overline{r} \gt -\overline{l}$当 0 <β<α0<α<α*); (i-2) 消失，即，$\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$和$\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$， 在哪里$\数学{T}$是某个正常数；(i-3) 过渡，即G(吨) → −∞,H(吨) → −∞,$0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$和你(吨, ⋅) →五(吨, ⋅), 其中五是一个吨- 具有紧凑支持的周期性解决方案。(ii) 万一β≥α0要么α≥α*，任何解决方案都会发生消失。