On a Multistable Type of Free Boundary Problem with a Flux at the Boundary

This paper studies the free boundary problem of a multistable equation with a Robin boundary condition, which may be used to describe the spreading of the invasive species with the solution representing the density of species and the free boundary representing the boundary of the spreading region. The Robin boundary condition $u_x\left(t,0\right)=\tau u\left(t,0\right)$ means that there is a flux of species at $x=0$ . By studying the asymptotic properties of the bounded solution, we obtain the following two situations: (i) four types of survival states: the solution is either big spreading (the solution converges to a big stationary solution defined on the half-line) or small spreading (the solution converges to a small stationary solution defined on the half-line) or small equilibrium state (the survival interval $\left[0,h\left(t\right)\right]$ tends to a finite interval and the solution tends to a small compactly supported solution) or vanishing happens (the solution and the interval $\left[0,h\left(t\right)\right]$ shrinks to 0 as $t⟶T$ for $T<+\infty$ ); (ii) a trichotomous survival states of solutions: big spreading, big equilibrium state, and vanishing.