Global continuation of monotone wavefronts

TL;DR

This paper establishes criteria for the existence of monotone traveling wavefronts in certain delayed reaction-diffusion equations, covering key models like Mackey-Glass and KPP-Fisher, using advanced functional-analytic methods.

Contribution

It provides a new theoretical framework for the global continuation of monotone wavefronts in non-quasi-monotone delayed reaction-diffusion equations.

Findings

Criteria for existence of monotone wavefronts established

Applicable to Mackey-Glass and KPP-Fisher equations

Uses Lyapunov-Schmidt reduction in weighted function spaces

Abstract

In this paper, we answer the question about the criteria of existence of monotone travelling fronts $u = \phi(\nu \cdot x+ct), \phi(-\infty) =0, \phi(+\infty) = \kappa,$ for the monostable (and, in general, non-quasi-monotone) delayed reaction-diffusion equations $u_t(t,x) - \Delta u(t,x) = f(u(t,x), u(t-h,x)).$ $C^{1,\gamma}$-smooth $f$ is supposed to satisfy $f(0,0) = f(\kappa,\kappa) =0$ together with other monostability restrictions. Our theory covers the two most important cases: Mackey-Glass type diffusive equations and KPP-Fisher type equations. The proofs are based on a variant of Hale-Lin functional-analytic approach to the heteroclinic solutions where Lyapunov-Schmidt reduction is realized in a `mobile' weighted space of $C^2$-smooth functions. This method requires a detailed analysis of a family of associated linear differential Fredholm operators: at this stage, the discrete Lyapunov functionals by Mallet-Paret and Sell are used in an essential way.