Uniqueness of fast travelling fronts in reaction-diffusion equations with delay

We consider positive travelling fronts, u(t, x) = phi(nu.x + ct), phi(-infinity) = 0, phi(infinity) = kappa, of the equation u(t)(t, x) = Delta u(t, x) - u(t, x) + g(u(t-h, x)), x is an element of R(m). This equation is assumed to have exactly two non-negative equilibria: u(1)equivalent to 0 and u(2)equivalent to k>0, but the birth function g is an element of C(2)(R, R) may be non-monotone on [0, k]. 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 phi(nu.x + ct) is unique (modulo translations). Note that f may be non-monotone. To prove uniqueness, we introduce a small parameter epsilon = 1/c and realize a Lyapunov Schmidt reduction in a scale of Banach spaces.