Elena Trofimchuk

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 reaction–diffusion equation of the monostable type.

Mackey-Glass type delay differential equations near the boundary of absolute stability

For an equation x � (t) =− x(t) + ζf (x(t − h)), x ∈ R, f � (0) =− 1, ζ> 0, with C 3 nonlinearity f which has a negative Schwarzian derivative and satisfies xf (x) 0a nd h(ζ − 1) 1/8 are less than some constant (independent on h, ζ ). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey–Glass type delay differential equations.  2002 Elsevier Science (USA). All rights reserved.

Traveling waves for a model of the Belousov–Zhabotinsky reaction

Abstract Following J.D. Murray, we consider a system of two differential equations that models traveling fronts in the Noyes-Field theory of the Belousov–Zhabotinsky (BZ) chemical reaction. We are also interested in the situation when the system incorporates a delay h ⩾ 0 . As we show, the BZ system has a dual character: it is monostable when its key parameter r ∈ ( 0 , 1 ] and it is bistable when r > 1 . For h = 0 , r ≠ 1 , and for each admissible wave speed, we prove the uniqueness of monotone wavefronts. Next, a concept of regular super-solutions is introduced as a main tool for generating new comparison solutions for the BZ system. This allows to improve all previously known upper estimations for the minimal speed of propagation in the BZ system, independently whether it is monostable, bistable, delayed or not. Special attention is given to the critical case r = 1 which to some extent resembles to the Zeldovich equation.

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).