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

• Published in: Journal of Differential Equations Vol. 259; no. 2; pp. 423 - 453
• Main Authors: Liu, Xiaowei, Lou, Bendong
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.07.2015
• Subjects: Free boundary condition; Long time behavior; Reaction–diffusion equation; Robin boundary condition; Sharp threshold; Robin boundary condition; Reaction–diffusion equation; Long time behavior; Free boundary condition; 35B20; Sharp threshold; 35R35; 35B40; 35K57
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2015.02.012

This paper studies the following problem{ut=uxx+f(u),0<x<h(t),t>0,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,∞), σ⩾0 and ϕ is a compactly supported C2 function. We prove that, there exists σ⁎>0 such that, vanishing happens when σ<σ⁎ (i.e., h(t)<M for some M>0 and u(⋅,t) converges as t→∞ to 0 uniformly in [0,h(t)]); spreading happens when σ>σ⁎ (i.e., h(t)−c⁎t tends to a constant for some c⁎>0, u(⋅,t) converges to a positive stationary solution locally uniformly in [0,∞) and to a traveling semi-wave with speed c⁎ near x=h(t)); in the transition case when σ=σ⁎, ‖u(⋅,t)−V(⋅−ξ(t))‖H2([0,h(t)]) tends to 0 as t→∞, where ξ(t) is a maximum point of u(⋅,t) and V is the unique even positive solution of V″+f(V)=0 subject to V(∞)=0. Moreover, with respect to b and f, ξ(t)=Pln⁡t+Q+o(1) for some P>0 and Q∈R, or, ξ(t)→z for some root z of V(−z)=bV′(−z).