On a reaction–diffusion equation with Robin and free boundary conditions

This paper studies the following problem{ut=uxx+f(u),00,u(0,t)=bux(0,t),t>0,u(h(t),t)=0,h′(t)=−ux(h(t),t),t>0,h(0)=h0,u(x,0)=σϕ(x),0⩽x⩽h0 where f   is an unbalanced bistable nonlinearity, b∈[0,∞)b∈[0,∞), σ⩾0σ⩾0 and ϕ   is a compactly supported C2C2 function. We prove that, there exists σ⁎>0σ⁎>0 such that, vanishing happens when σ<σ⁎σ<σ⁎ (i.e., h(t)0M>0 and u(⋅,t)u(⋅,t) converges as t→∞t→∞ to 0 uniformly in [0,h(t)][0,h(t)]); spreading happens when σ>σ⁎σ>σ⁎ (i.e., h(t)−c⁎th(t)−c⁎t tends to a constant for some c⁎>0c⁎>0, u(⋅,t)u(⋅,t) converges to a positive stationary solution locally uniformly in [0,∞)[0,∞) and to a traveling semi-wave with speed c⁎c⁎ near x=h(t)x=h(t)); in the transition case when σ=σ⁎σ=σ⁎, ‖u(⋅,t)−V(⋅−ξ(t))‖H2([0,h(t)])‖u(⋅,t)−V(⋅−ξ(t))‖H2([0,h(t)]) tends to 0 as t→∞t→∞, where ξ(t)ξ(t) is a maximum point of u(⋅,t)u(⋅,t) and V   is the unique even positive solution of V″+f(V)=0V″+f(V)=0 subject to V(∞)=0V(∞)=0. Moreover, with respect to b and f  , ξ(t)=Pln⁡t+Q+o(1)ξ(t)=Pln⁡t+Q+o(1) for some P>0P>0 and Q∈RQ∈R, or, ξ(t)→zξ(t)→z for some root z   of V(−z)=bV′(−z)V(−z)=bV′(−z).