## 9th session, March 16th & 17th 2015

__Monday 16th__

*EHESS, 190-198*

*Avenue de France, room 466, 4th floor - 75013 Paris*

**

14h30 : Hiroshi Matano

*(Professor at Graduate School of Mathematical Sciences, Université de Tokyo, Professeur invité à l’EHESS / chaire ReaDi)*
Spreading fronts in nonlinear diffusion equations, Part 1 : Radially symmetric terraceIn these two talks I discuss the long time behavior of solutions of a nonlinear diffusion equations on R^N with compactly supported intial data. I mainly focus on the question of how the solution fronts spread over the space. In Part 1, I will consider semilinear diffusion equations with rather general -- possibly multi-stable -- nonlinearities satisifying f(0) = 0, f'(0) < 0, and show that solution fronts approaches what we call a "radially symmetric terrace". This is joint work with Yihong Du. |

15h30 : Alessandro Zilio

*(Post-doc ReaDi)*

Theoretical results in phase separation models: an approach based on systems of elliptic equationsIn this seminar I will talk about some results concerning systems of elliptic equations which appear in the context of phase separation models, giving a brief review of the literature and presenting some recent progresses. More precisely, I will comment on some qualitative properties, namely the regularity in Hœlder spaces, which are shared by the solutions of such systems and the dependence of these properties on the parameters governing the phase separation phenomenon. The precise knowledge of the behaviour of the solutions will then be used in order to derive existence results, clarifying the picture of the separation of phases. |

__Tuesday 17th__

*EHESS, 190-198*

*Avenue de France, room 466, 4th floor - 75013 Paris*

**

10h : Hiroshi Matano

*(Professor at Graduate School of Mathematical Sciences, Université de Tokyo, Professeur invité à l’EHESS / chaire ReaDi)*
Spreading fronts in nonlinear diffusion equations, Part 2 : Anisotropic Allen-Cahn equation and Wulff shapeIn these two talks I discuss the long time behavior of solutions of a nonlinear diffusion equations on R^N with compactly supported intial data. I mainly focus on the question of how the solution fronts spread over the space. In Part 2, I will consider an anisotropic Allen-Cahn equation on R^N. This is a generalization of the classical Allen-Cahn equation and has an anisotropic diffusion term which is highly nonlinear. We show that the shape of the spreading front converges to the Wulff shape associated with the anisotropic term of the equation. This is joint work with Yoichiro Mori and Mitsunori Nara. |

Comportement asymptotique de l'équation de Fisher-KPP multi-dimensionnelleDans cet exposé, je présenterai un résultat de convergence vers un front d'onde pour une équation de Fisher-KPP en dimension quelconque avec une donnée initiale à support compact. De façon plus précise, l'équation de KPP est posé dans milieux hétérogène en espace mais asymptotiquement homogène. On montrera que la position du front dépend fortement de la vitesse de convergence du milieux à l'infini. Une convergence suffisamment rapide ne change pas la position du front par rapport au cas homogène alors qu'une convergence lente décale le front avec un écart tendant vers l'infini. |

15h : Hiroshi Matano

*(Professor at Graduate School of Mathematical Sciences, Université de Tokyo, Professeur invité à l’EHESS / chaire ReaDi)*

Dans le cadre du séminaire du CAMS La question de la modélisation en sciences humaines
Attention au lieu : Salle 1, EHESS, 105 boulevard Raspail, 75006 – Paris Order-preserving dynamical systems with mass conservationThe theory of order-preserving dynamical systems was largely developed in 1980's and 90's after the pioneering work of M.W. Hirsch and others. What is remarkable about this theory is that it allows us to derive various important qualitative properties of solutions -- such as stability and convergence -- solely by a slightly stronger version of the usual comparison principle, without further knowledge about the specific features of the equations. Recently there have been some new develpments in this theory. In this talk I will present our new results on order-preserving systems with a mass conservation property (or a first integral). Our results extend the earlier work by Arino (1991), Mierczynski (1995, 2012) and Banaji-Angeli (2010) considerably with a significantly simpler proof. I will then apply this theory to a number of problems including mathematical models for transportation by molecular motors , chemical reservible reactions, competition-diffusion systems, and so on. This is joint work with Toshiko Ogiwara and Danielle Hilhorst. |