Nonlinearly determined wavefronts of the Nicholson's diffusive equation: when small delays are not harmless

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:268

Article

Chladna, Zuzana¹  Hasik, Karel²  Kopfova, Jana²  Nabelkova, Petra²  Trofimchuk, Sergei³ 

[1] Comenius Univ, Fac Math Phys & Informat, Dept Appl Math & Stat, Bratislava 84248, Slovakia

[2] Silesian Univ, Math Inst, Opava 74601, Czech Republic

[3] Univ Talca, Inst Matemat & Fis, Casilla 747, Talca, Chile

关键词: Non-linear determinacy;    Delay;    Wavefront;    Existence;    Super-exponential solution;   

DOI  :  10.1016/j.jde.2019.11.007

来源: Elsevier

【 摘 要 】

By proving the existence of non-monotone and non-oscillating wavefronts for the Nicholson's blowflies diffusive equation (the NDE), we answer an open question from [16]. Surprisingly, wavefronts of such a kind can be observed even for arbitrarily small delays. Similarly to the pushed fronts, obtained waves are not linearly determined. In contrast, a broader family of eventually monotone wavefronts for the NDE is indeed determined by properties of the spectra of the linearized equations. Our proofs use essentially several specific characteristics of the blowflies birth function (its unimodal form and the negativity of its Schwarz derivative, among others). One of the key auxiliary results of the paper shows that the Mallet-Paret-Cao-Arino theory of super-exponential solutions for scalar equations can be extended for some classes of second order delay differential equations. For the new type of non-monotone waves to the NDE, our numerical simulations also confirm their stability properties established by Mei et al. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free