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

Abstract

In Liu and Lou (2015), the authors considered the reaction–diffusion equation u t = u x x + f ( u ) with Robin and free boundary conditions. For the initial data σ ϕ , 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 ) ln t + O ( 1 ) in Case 1 and h ( t ) = 3 2 − f ′ ( 0 ) ln t + O ( 1 ) in Case 2.