Min-Max formulae for the speeds of pulsating travelling fronts in periodic excitable media

TL;DR

This paper derives min-max and max-min formulas for the speeds of pulsating travelling fronts in periodic reaction-advection-diffusion media, providing new variational characterizations for combustion and KPP nonlinearities.

Contribution

It introduces novel min-max and max-min formulas for propagation speeds in periodic media, including variational formulas involving eigenvalue problems.

Findings

Unique speed c for combustion nonlinearities with min-max formula

Existence of minimal speed c* for KPP and ZFK nonlinearities with min-max formula

Application of formulas to eigenvalue problems for minimal speed c*

Abstract

This paper is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equations in a periodic framework. It deals with travelling wave solutions of the equation $$u_t =\nabla\cdot(A(z)\nabla u) +q(z)\cdot\nabla u+ f(z,u), t \in \mathbb{R}, z \in \Omega,$$ propagating with a speed $c.$ In the case of a "combustion" nonlinearity, the speed $c$ exists and it is unique, while the front $u$ is unique up to a translation in $t.$ We give a $\min-\max$ and a $\max-\min$ formula for this speed $c.$ On the other hand, in the case of a "ZFK" or a "KPP" nonlinearity, there exists a minimal speed of propagation $c^{*}.$ In this situation, we give a $\min-\max$ formula for $c^{*}.$ Finally, we apply this $\min-\max$ formula to prove a variational formula involving eigenvalue problems for the minimal speed $c^{*}$ in the "KPP" case.