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

Abstract

We consider positive travelling fronts, u ( t , x )= ϕ ( ν . x + ct ), ϕ (−∞)=0, ϕ (∞)= κ , of the equation u t ( t , x )=Δ u ( t , x )− u ( t , x )+ g ( u ( t − h , x )), x ∈ m . This equation is assumed to have exactly two non-negative equilibria: u 1 ≡0 and u 2 ≡ κ >0, but the birth function g ∈ C 2 ( , ) 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 ϵ =1/ c and realize a Lyapunov–Schmidt reduction in a scale of Banach spaces.

Uniqueness of non-monotone traveling waves for delayed reaction-diffusion equations

“…(1.2) have been considered by many researchers; see, e.g., [2,3,8,9]. For the case where f is non-monotone on [0, K ], i.e., p/δ > e, there are a few results; see [18,10,13,17]. Here, we consider the case for e < p/δ ≤ e 2 .…”

On the geometry of wave solutions of a delayed reaction–diffusion equation

“…(1) is separated from zero as x + ct → +∞. The persistence of semi-wavefronts was established in [32] for a local version of model (1). The proof in [32] is based on the local estimations technique which does not apply to Eq.…”

Uniqueness of critical traveling waves for nonlocal lattice equations with delays

“…To my knowledge, no result on uniqueness of critical traveling waves for delayed lattice systems has been reported. Other methods to prove uniqueness of noncritical traveling waves for other types of evolution systems can be found in [1,3,4,5,6,7,11,12]. In [8], it is also shown that for (1.1), the minimal wave speed c * coincides with the spreading speed and the linear determinacy holds for (1.1), meaning that c * is fully determined by the characteristic equation of the linearization of (1.1) at the trivial equilibrium.…”