Spreading in advective environment modeled by a reaction diffusion equation with free boundaries

We consider a reaction–diffusion–advection equation of the form: ut=uxx−βux+f(u)ut=uxx−βux+f(u) for x∈(g(t),h(t))x∈(g(t),h(t)), where g(t)g(t) and h(t)h(t) are two free boundaries satisfying Stefan conditions, f is a bistable type of nonlinearity. This equation is used to describe the population dynamics in advective environment. We study the influence of the advection coefficient β   on the dynamics of the solutions. We find a parameter β⁎>c0β⁎>c0 (where c0>0c0>0 is the speed of the traveling wave of the equation ut=uxx+f(u)ut=uxx+f(u) ) such that when β∈(0,c0)β∈(0,c0) (resp. when β∈(c0,β⁎)β∈(c0,β⁎)), there is a vanishing-transition-spreading (resp. vanishing-transition-virtual spreading) trichotomy result for the long time behavior of the solutions; when β⩾β⁎β⩾β⁎, all the solutions vanish (i.e., h(t)−g(t)h(t)−g(t) is bounded and u→0u→0 uniformly).