A reaction-diffusion-advection equation with mixed and free boundary conditions

We investigate a reaction-diffusion-advection equation of the form $u_t-u_{xx}+\beta u_x=f(u)$ $(t>0,\,0<x<h(t))$ with mixed boundary condition at $x=0$ and a free boundary condition at $x=h(t)$. Such a model may be applied to describe the dynamical process of a new or invasive species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundary representing the expanding front. The goal of this paper is to understand the effect of advection environment and no flux across the left boundary on the dynamics of this species. When $|\beta|<c_0$, we first derive the spreading-vanishing dichotomy and sharp threshold for spreading and vanishing. Then provide a much sharper estimate for the spreading speed of $h(t)$ and the uniform convergence of $u(t,x)$ when spreading happens. For the case $|\beta|\geq c_0$, some results concerning spreading, virtual spreading, vanishing and virtual vanishing are obtained. Where $c_0$ is the minimal speed of traveling waves of the differential equation.