Spreading speeds and uniqueness of traveling waves for a reaction diffusion equation with spatio-temporal delays

Xu, Zhaoquan; Xiao, Dongmei

doi:10.1016/j.jde.2015.08.049

Cited by 25 publications

(28 citation statements)

References 45 publications

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“…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]…”

Section: Introduction and Main Resultssupporting

confidence: 73%

“…In consequence, different strategies have been elaborated so far to tackle the uniqueness problem, e.g. see [1,2,4,5,6,7,8,9,11,12,14,18,21,24,26,27,28,31,33]. Broadly speaking, the cited works show that the wave uniqueness can be established when either the evolution equation or the waves are monotone, or when the Lipschitz constant of nonlinear reaction term is dominated by its derivative at the unstable equilibrium (the Diekmann-Kaper condition).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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A simple approach to the wave uniqueness problem

Solar

Trofimchuk

2019

Journal of Differential Equations

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We propose a new approach for proving uniqueness of semi-wavefronts in generally nonmonotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.

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“…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]…”

Section: Introduction and Main Resultssupporting

confidence: 73%

Section: Introduction and Main Resultsmentioning

confidence: 99%

A simple approach to the wave uniqueness problem

Solar

Trofimchuk

2019

Journal of Differential Equations

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“…We note that the bulk of information concerning the Nicholson's diffusive equation is obtained for a simpler (monotone) case when 1 < p/δ ≤ e (cf. the recent works [48] and [39]).…”

Section: Main Assumptionsmentioning

confidence: 91%

“…In general, c * (h) is not linearly determined, i.e. C(h) = [c # (h), +∞) [23,48]. The minimal wave is called pushed if c * (h) > c # (h).…”

Section: Main Assumptionsmentioning

confidence: 99%

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

Trofimchuk¹,

Pinto

Trofimchuk

2016

Journal of Differential Equations

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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

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“…The existence, uniqueness and asymptotic behavior of the non-monotone traveling waves of (1.1) have been guaranteed by [18]. Our study includes the Mei's [1] work.…”

Section: Introductionmentioning

confidence: 99%