Monotone waves for non-monotone and non-local monostable reaction–diffusion equations

Abstract

We propose a new approach for proving existence of monotone wavefronts in non-monotone and non local monostable diffusive equations. This allows to extend recent results established for the particular case of equations with local delayed reaction. In addition, we demonstrate the uniqueness (modulo translations) of obtained monotone wavefront within the class of all monotone wavefronts (such a kind of conditional uniqueness was recently established for the non-local KPP-Fisher equation by Fang and Zhao). Moreover, we show that if delayed reaction is local then each monotone wavefront is unique (modulo translations) within the class of all non-constant traveling waves. Our approach is based on the construction of suitable fundamental solutions for linear integral-differential equations. We consider two alternative scenarios: in the first one, the fundamental solution is negative (typically holds for the Mackey-Glass diffusive equations) while in the second one, the fundamental solution is non-negative (typically holds for the KPP-Fisher diffusive equations).FONDECYT (Chile) 1110309 1120709 115048

A simple approach to the wave uniqueness problem

“…A similar situation is also observed for another popular delayed model, the Mackey-Glass type diffusive equation [1,3,11,21,24,26,28,29,31,33,34]…”

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

“…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.…”

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

“…Then we observe that the theory developed in [21] for the Lebesgue integrable kernel K(s) applies literally to the case of the generalized kernels. In particular, in view of Lemma 5 above, [21,Lemma 19] holds with ξ * ≥ 1. Observe that we can admit d = 0 in [21, Lemma 19] because of the following property of the above given kernel…”

“…so that, z 0 τ = −2(1 + 1 + 4b/τ) −1 =: −σ, σ ∈ (0, 1).Hence, we have obtained a parametric solution of(21): τ = σ 2 e −σ , b = (1 − σ)e −σ , σ ∈ (0, 1). Note that τ ′ (b) = −σ < 0, other properties of τ (b) are also obvious.…”

“…Obviously, Wu and Zou iteration method can be applied only if the differential equation itself possesses monotone wavefronts. As the studies [5,21] show, often this happens if and only if the nonlinearity of equation displays some kind of quasi-monotonicity. Accordingly, [27] considers two different types of the quasi-monotonicity property for the monostable delayed model system…”