Nonstandard Quasi-monotonicity: An Application to the Wave Existence in a Neutral KPP–Fisher Equation

Abstract

We revisit Wu and Zou non-standard quasi-monotonicity approach for proving existence of monotone wavefronts in monostable reaction-diffusion equations with delays. This allows to solve the problem of existence of monotone wavefronts in a neutral KPP-Fisher equation. In addition, using some new ideas proposed recently by Solar et al., we establish the uniqueness (up to a translation) of these monotone wavefronts.

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

“…Our subsequent analysis is inspired by the arguments proposed in [7] and [18], we present them here for the sake of completeness. Set y(t) = 1 − φ(t), where φ(t) is a positive monotone wavefront.…”

“…In view of (18), this implies that, for all t ∈ R, γ). Therefore, for some C > 0 and σ = 2r −1 0 lnν < 0,…”

“…On the other hand, derivation of explicit lower estimates for them seems to be rather problematic. At the same time, available upper estimations of τ (c) do not exclude that it may happen that τ (c) → 0 as c → ∞, see [14,29] or related argumentation in [18]. Certainly, all this limits the applicability of Proposition 1.…”

“…In our analysis of the wavefront existence problem for the FL model, we are using an appropriate modification of the Zou and Wu monotone iterations algorithm from [32]. The recent works [7,11,18,27] have shown high efficiency of this method in the studies of delayed [11,27], nonlocal [7] and neutral [18] versions of the KPP-Fisher equation as well as of the nonlocal diffusive equation of the Mackey-Glass type [27]. In this regard, this article provides a natural extension, for γ > 0, of the findings in [7,11] containing them as very particular cases.…”

Traveling wave solutions for a neutral reaction–diffusion equation with non-monotone reaction

“…In the last few decades, neutral equations have been investigated wildly, and a variety of themes have been touched, such as the existence and uniqueness, regularity and stability of solutions [1,4,9,11,14,16,28], periodic solutions [10,24], controllability [23], and so on. But, to the best of our knowledge, there are few works that treat the traveling wave solution of partial neutral functional equations [12,13,20].…”