Abstract

We consider positive travelling fronts, u(t,x)=φ(ν•x+ct), φ(-∞)=0, φ(∞)=κ, of the equation ut(t,x)=Δu(t,x)-u(t,x)+g(u(t-h,x)), xεR^m. This equation is assumed to have exactly two non-negative equilibria: u1=≡0 and u2= ≡κ>0, but the birth function gεC²(R, R) may be non-monotone on [0,κ]. We are therefore interested in the so-called monostable case of the time-delayed reaction-diffusion equation. Our main result shows that for every fixed and sufficiently large velocity c, the positive travelling front φ(ν•x+ct) is unique (modulo translations). Note that φ may be non-monotone. To prove uniqueness, we introduce a small parameter =1/c and realize a Lyapunov-Schmidt reduction in a scale of Banach spaces. © 2008 The Royal Society.