Global continuation of monotone wavefronts

Abstract

In this paper, we establish efficient existence criteria for monotone traveling fronts u = ϕ(ν · x + ct), ϕ(−∞) = 0, ϕ(+∞) = κ of the monostable (and, in general, nonquasi-monotone) delayed reaction–diffusion equations ut(t, x) − Δu(t, x) = f(u(t, x), u(t − h, x)). The function f is of class C1,γ and it is assumed to satisfy f(0, 0) = f(κ, κ) = 0 together with other monostability restrictions. Our theory covers several important cases including Mackey–Glass-type diffusive equations and Kolmogorov-Petrovskii-Piskunov-Fisher-type equations. The proofs are based on a variant of the Hale–Lin functional-analytic approach to heteroclinic solutions where Lyapunov–Schmidt reduction is realized in a 'mobile' weighted space of C2-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 essential to the method.