Convergence of solutions for Fisher–KPP equation with a free boundary condition

Abstract

We consider a free boundary problem: $u_t=u_{xx}+f\left(t,u\right)$ ($g\left(t\right)<x<h\left(t\right)$) with free boundary conditions $h^{\prime }\left(t\right)=-u_x\left(t,h\left(t\right)\right)-\alpha \left(t\right)$ and $g^{\prime }\left(t\right)=-u_x\left(t,g\left(t\right)\right)+\alpha \left(t\right)$, where $\alpha \left(t\right)$ is a positive $T$-periodic function, $f\left(t,u\right)$ is a Fisher–KPP type of nonlinearity and $T$-periodic in $t$. Such a problem can model the spreading of a biological or chemical species in time-periodic environment, where free boundaries mimic the spreading fronts of the species. We mainly study the convergence of bounded solutions. There is a $T$-periodic function ${\alpha }_0\left(t\right)$ which plays a key role in the dynamics. More precisely:

(i) When $0<\alpha <{\alpha }_0$, we obtain a trichotomy result:

1. Spreading, i.e., $h\left(t\right),-g\left(t\right)\to +\infty$ and $u\left(t,\cdot \right)\to P\left(t\right)$ as $t\to \infty$, where $P\left(t\right)$ is the periodic solution of the ODE $u_t=f\left(t,u\right)$.

2. Vanishing, i.e., $\underset{t\to \mathcal{𝒯}}{lim}h\left(t\right)=\underset{t\to \mathcal{𝒯}}{lim}g\left(t\right)$ and $\underset{t\to \mathcal{𝒯}}{lim}\underset{g\left(t\right)\le x\le h\left(t\right)}{max}u\left(t,x\right)=0$, where $\mathcal{𝒯}$ is some positive constant.

3. Transition, i.e., $0<\underset{t\to \infty }{lim}h\left(t\right),\underset{t\to \infty }{lim}g\left(t\right)<+\infty$, $0<\underset{t\to \infty }{lim}\left[h\left(t\right)-g\left(t\right)\right]<+\infty$ and $u\left(t,\cdot \right)\to V\left(t,\cdot \right)$, where $V$ is a $T$-periodic solution with compact support.

(ii) In the case $\alpha \ge {\alpha }_0$, vanishing happens for any solution.