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

Source abstract

We study a free boundary problem of the form: u t = u xx + f ( t , u ) ( g ( t ) &lt; x &lt; 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 &lt; α 0 &lt; α * which play key roles in the dynamics. More precisely, (i) in case 0 &lt; β &lt; α 0 and 0 &lt; α &lt; 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 ) &gt; 0 &gt; − l ( t ) when 0 &lt; β &lt; α &lt; α 0 ; r ( t ) = 0 &gt; − l ( t ) when 0 &lt; β &lt; α = α 0 ; $0 \gt \overline{r} \gt -\overline{l}$ when 0 &lt; β &lt; α 0 &lt; α &lt; 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 ≥ α 0 or α ≥ α *, vanishing happens for any solution.