Abstract

We study positive bounded wave solutions u (t, x) = φ{symbol} (ν ṡ x + c t), φ{symbol} (- ∞) = 0, of equation ut (t, x) = Δ u (t, x) - u (t, x) + g (u (t - h, x)), x ∈ Rm (*). This equation is assumed to have two non-negative equilibria: u1 ≡ 0 and u2 ≡ κ > 0. The birth function g ∈ C (R+, R+) is unimodal and differentiable at 0 and κ. Some results also require the feedback condition (g (s) - κ) (s - κ) < 0, with s ∈ [g (max g), max g] {set minus} {κ}. If additionally φ{symbol} (+ ∞) = κ, the above wave solution u (t, x) is called a travelling front. We prove that every wave φ{symbol} (ν ṡ x + c t) is eventually monotone or slowly oscillating about κ. Furthermore, we indicate c* ∈ R+ ∪ {+ ∞} such that Eq. (*) does not have any travelling front (neither monotone nor non-monotone) propagating at velocity c > c*. Our results are based on a detailed geometric description of the wave profile φ{symbol}. In particular, the monotonicity of its leading edge is established. We also discuss the uniqueness problem indicating a subclass G of 'asymmetric' tent maps such that given g ∈ G, there exists exactly one positive travelling front for each fixed admissible speed. © 2008 Elsevier Inc. All rights reserved.