|Cette journée de colloque transfrontalier, organisée par la Fédération de Mathématique Rhône-Alpes Auvergne (FMRAA), est l'occasion de rassembler des chercheurs français, italiens et suisses autour de la thématique des fronts de propagation ou traveling waves et leurs applications en physique, chimie ou écologie. Cinq exposés pléniers de 50 minutes chacun seront proposés au cours de cette journée. Nous clôturerons cette journée par une période de discussion pour faciliter la prise de contact et les collaborations entre chercheurs.|
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We are concerned with transition fronts for reaction-diffusion equations of the Fisher-KPP type. Basic examples are the standard traveling fronts, but the class of transition fronts is much larger. We will describe it and study its qualitative dynamical properties in the case of homogeneous equations. In particular, we characterize the set of admissible asymptotic past and future speeds, as well as the asymptotic profiles, and we show that transition fronts can only accelerate. In the second part of the talk, we will apply these results to describe the transition fronts in the case of a time-dependent nonlinear term admitting two different limits in the past and in the future. This is a joint work with Francois Hamel.
Romain Joly (Université Joseph Fourier)
Several classical PDEs admit a (at least formal) Lyapounov function, that is an energy decaying along the trajectories of the PDE. But some of them have a more special structure: for any speed c, if one considers the PDE in the Galilean frame moving at speed c, then this new PDE still admits a (at least formal) Lyapounov function. We will see that this structure enables us to prove existence and global stability of traveling fronts. In particular, these technics apply to systems of parabolic PDEs, or to the damped wave equation, where no comparaison principle is available. This talk is based on a joint work with Thierry Gallay and a previous work of Emmanuel Risler.
I will discuss front propagation problems for forced mean curvature flow and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. I will present a convergence result relating asymptotic in time front propagation in the di ffuse interface case to that in the sharp interface case for suitably balanced nonlinearities of Allen-Cahn type. The result is obtained by means of a variational approach.
Emeric Bouin (École Normale Supérieure de Lyon)
Recently, kinetic-type models have raised a lot of interest in multiscale modelling of collective motion and dispersal evolution. For instance, kinetic models are a very good option for modelling concentration waves of chemotactic bacteria in a micro-channel. Another example is the propagation of some invasive species with a high heterogeneity in dispersal capability among individuals. A minimal reaction-diffusion model, very similar to a kinetic equation, has recently been proposed by Benichou et al. I will present some recent progresses about the existence (and non-existence) of travelling waves for two analogous models. First, a kinetic reaction-transport model very close to the Fisher-KPP equation. Then, a reaction-diffusion model where the phenotypical heterogeneity in the population affects the diffusion of individuals.
Vitaly Volpert (Université Lyon 1)