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

Abstract

The aim of this paper is to study the existence and the geometry of positive bounded wave solutions to a non-local delayed reactiondiffusion equation of the monostable type.

Asymptotic stability of traveling waves for delayed reaction-diffusion equations with crossing-monostability

“…For the sake of convenience, we first introduce the results of Wu et al [ ] on the existence and uniqueness of the non-monotone traveling waves of (1.1) in the crossing-monostable case (see also Ma [17], Faria and Trofimchuk [6] and Trofimchuk et al [28]). Assume that…”

“…2), the traveling wave problem becomes harder, especially for the stability of traveling waves, due to the lack of quasimonotonicity. Recently, there have been many efforts on the existence of traveling waves for the nonmonotone delayed equations, see e.g., [5,6,25,28,30,34], in which these authors showed the existence for sufficiently large wave speeds or small delays. Ma [17] further applied Schauder's fixed-point theorem to a delayed non-local reaction-diffusion equation and obtained the existence of traveling waves for all values of the time delay.…”

On the minimal speed of traveling waves for a nonlocal delayed reaction–diffusion equation

“…(1.1) at the trivial steady state. From [4,7], we know that there is ε 0 = ε 0 (h) > 0 such that ψ(z, ε 0 ) = 0 has a unique multiple positive root z 0 = z 0 (h). Furthermore, if g(s) ≤ g (0)s for s ≥ 0, then the minimal speed c * is equal to c * = 1/ √ ε 0 .…”

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

“…For the sake of convenience, we first introduce the result of [13, Theorem 1.2] on the existence of non-monotone traveling waves of (1.1) (see also Ma [12], Faria and Trofimchuk [11] and Trofimchuk et al [17]). Assume that there exists K *…”

“…(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 .…”