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

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 (β = 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⁎ ≥C, and the long-time behavior of the species is determined by trichotomy when the shifting speed c⁎ ∈(0,C). Here we mainly consider the problems with advection and shifting speed c⁎ ∈(0,C) (the case c⁎ ≥ C can be studied by similar methods in this paper). We prove that there exist β* <0 and β⁎ >0 such that the species always dies out in the long-run when β ≤ β*, while for β ∈(β*,β⁎) or β = β⁎, the long-time behavior of the species is determined by the corresponding trichotomies respectively.