Global continuation of monotone wavefronts

Abstract

In this paper, we answer the question about the criteria of existence of monotone travelling fronts $u=\varphi (\nu \cdot x+ct),\varphi (-\infty )=0,\varphi (+\infty )=\kappa ,$ 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.

21 pages, 3 figures, submitted

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