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

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 $c^{\ast }.$ In this situation, we give a $\min -\max$ formula for $c^{\ast }.$ Finally, we apply this $\min -\max$ formula to prove a variational formula involving eigenvalue problems for the minimal speed $c^{\ast }$ in the "KPP" case.