The transition speed of reaction-diffusion problems with Robin and free boundary conditions

In Liu and Lou (2015), the authors considered the reaction-diffusion equation ut=uxx+f(u) with Robin and free boundary conditions. For the initial data σϕ, there exists σ∗>0 such that spreading happens when σ>σ∗ and vanishing happens when σ<σ∗. In the transition case that σ=σ∗, the solution u(t,x) converges to the ground state with a suitable shift: V(⋅−ξ(t)), and ξ(t) tends to a finite number (Case 1) or to ∞ (Case 2) as t→∞. For both cases, the right free boundary h(t) always propagate to infinity. In this paper, we will discuss the expanding speed of h(t) of these two cases. Actually, h(t)=1−f′(0)lnt+O(1) in Case 1 and h(t)=32−f′(0)lnt+O(1) in Case 2.