Min-Max formul{æ} for the speeds of pulsating travelling fronts in periodic excitable media

Mohammad El Smaily (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

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.

Domaines: Equations aux dérivées partielles [math.AP]

Mohammad El Smaily. Min-Max formul{æ} for the speeds of pulsating travelling fronts in periodic excitable media. 2008. ⟨hal-00283500⟩