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, Wei and Zhou in [16] 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 shifing 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.