2nd session, November 12 & 13th, 2013
Video report here
Tuesday 12
9h30 - 10h : Coffee
10h - 11h : François Hamel (LATP, Aix-Marseille)
Bistable transition fronts in R^N
The standard notions of reaction-diffusion waves or fronts can be viewed as examples of generalized transition waves. These notions, introduced by H. Berestycki and F. Hamel, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. The existence of transition waves has been proved in various contexts where the standard notions of waves make no longer sense. Even for homogeneous equations in R^N, fronts with various non-planar shapes are known to exist. In these talks, I will also report on some qualitative properties of bistable transition fronts in RN and in particular on the uniqueness of their global mean speed, regardless of their shape. I will also mention the existence of transition fronts in R^N which are not standard traveling fronts.
The standard notions of reaction-diffusion waves or fronts can be viewed as examples of generalized transition waves. These notions, introduced by H. Berestycki and F. Hamel, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. The existence of transition waves has been proved in various contexts where the standard notions of waves make no longer sense. Even for homogeneous equations in R^N, fronts with various non-planar shapes are known to exist. In these talks, I will also report on some qualitative properties of bistable transition fronts in RN and in particular on the uniqueness of their global mean speed, regardless of their shape. I will also mention the existence of transition fronts in R^N which are not standard traveling fronts.
11h - 12h : Alessandro Zilio (Politecnico di Milano)
Strongly competing systems with fractional diffusion: uniform regularity results
In this talk I will focus on some recent results obtained in collaboration with S. Terracini and G. Verzini about the optimal regularity of solution to systems characterized by nonstandard diffusion and strong competition. In particular, it will be shown that the type of competition strongly affects both the regularity of the solutions and the geometry of the segregated states obtained in the limiting case of infinite competition.Cliquez ici pour modifier.
In this talk I will focus on some recent results obtained in collaboration with S. Terracini and G. Verzini about the optimal regularity of solution to systems characterized by nonstandard diffusion and strong competition. In particular, it will be shown that the type of competition strongly affects both the regularity of the solutions and the geometry of the segregated states obtained in the limiting case of infinite competition.Cliquez ici pour modifier.
15h-16h : Amandine Aftalion (Directrice de recherche au CNRS, Université de Versailles Saint-Quentin)
Questions de modélisation dans le sport : existe-t-il une stratégie de course optimale?
Séminaire du CAMS « La question de la modélisation des sciences humaines : mathématiques et informatique ». voir ici
Séminaire du CAMS « La question de la modélisation des sciences humaines : mathématiques et informatique ». voir ici
16h-17h : discussion
Wednesday 13
9h30 - 10h : Coffee
Suite de l’exposé du 12 novembre
11h -12h : Guillemette Chapuisat (LATP, Aix-Marseille)
This talk deals with the existence of traveling fronts for reaction-diffusion equations arising in biology and medicine where the reaction term is heterogeneous in space. Precisely I will focus on reaction-diffusion equations set on R^N with an area where the reaction term behaves like a KPP or bistable nonlinearity and an area where the reaction term is negative. In a first part, I will present various models coming from biology and medicine that involve such type of equations and study the existence of traveling front numerically. Then in a second part, I will sketch the proof of the existence theorem. It is based on the sliding method as in \textit{Travelling fronts in cylinders} (Berestycki & Nirenberg,1992) but the proof can directly be applied for a KPP nonlinearity by building appropriate sub- and super-solutions.