Long time behavior for solutions of the diffusive logistic equation with advection and free boundary

Abstract

We consider the influence of a shifting environment and an advection on the spreading of an invasive species through a model given by the diffusive logistic equation with a free boundary. When the environment is shifting and without advection ($\beta =0$), Du et al. (Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary. arXiv:1508.06246, 2015) showed that the species always dies out when the shifting speed $c_*\ge \mathcal {C}$, and the long-time behavior of the species is determined by trichotomy when the shifting speed $c_*\in (0,\mathcal {C})$. Here we mainly consider the problems with advection and shifting speed $c_*\in (0,\mathcal {C})$ (the case $c_*\ge \mathcal {C}$ can be studied by similar methods in this paper). We prove that there exist $\beta ^*<0$ and $\beta _*>0$ such that the species always dies out in the long-run when $\beta \le \beta ^*$, while for $\beta \in (\beta ^*,\beta _*)$ or $\beta =\beta _*$, the long-time behavior of the species is determined by the corresponding trichotomies respectively.