1. Danielle Hilhorst 氏： "A multiple scale pattern formation cascade in reaction-diffusion systems of activator-inhibitor type"

Abstract： A family of singular limits of reaction-diffusion systems of activator-inhibitor type in which stable stationary sharp-interface patterns may form is investigated. For concreteness, the analysis is performed for the FitzHugh-Nagumo model on a suitably rescaled bounded domain in $\R^N$, with $N \geq 2$. It is proved that when the system is sufficiently close to the limit the dynamics starting from the appropriate smooth initial data breaks down into five distinct stages on well-separated time scales, each of which can be approximated by a suitable reduced problem. The analysis allows to follow fully the progressive refinement of spatiotemporal patterns forming in the systems under consideration and provides a framework for understanding the pattern formation scenarios in a large class of physical, chemical, and biological systems modeled by the class of reaction-diffusion equations, which we consider.
This is joint work with Marie Henry and Cyrill Muratov.

2. Thanh Nam Nguyen 氏： "Formal asymptotic limit of a diffuse interface tumor-growth model"

Abstract： We consider a diffuse interface tumor-growth model, which has the form of a phase-field system. We discuss the singular limit of this problem. More precisely, we formally prove that as the reaction coefficient tends to zero, the solution converges to the solution of a free boundary problem.
This is a joint work with Danielle Hilhorst, Johannes Kampmann and Kristoffer G. van der Zee.

3. Peter Gordon 氏： "Gelfand type problem for two phase porous media"

Abstract： In this talk I will introduce a generalization of well known Gelfand problem arising in a Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two phase materials, where, in contrast to classical Gelfand problem which utilizes single temperature approach, the state of the system is described by two different temperatures. As a result the problem is modeled by a system of two coupled nonlinear heat equations. The new ingredient in such a generalized Gelfand problem is a presence of inter-phase heat exchange which can be viewed as a strength of coupling for the system.
I will show that similar to classical Gelfand problem the thermal explosion (blow up of solution) for generalized Gelfand problem occurs exclusively due to the absence of stationary temperature distribution, that is non-existence of solution of corresponding elliptic problem. I also will show that the presence of inter-phase heat exchange delays a thermal explosion. Moreover, in the limit of infinite heat exchange between phases the problem of thermal explosion in two phase porous media reduces to classical Gelfand problem with re-normalized constants. The latter result partially justifies a single temperature approach to two phase systems often used in a physical literature.
This is a joint work with Vitaly Moroz (Swansea University).

4. Cyrill Muratov 氏： "On the shape of charged drops: an isoperimetric problem with a competing non-local term"

Abstract： In this talk I will give an overview of my recent work with H. Knuepfer on the analysis of a class of geometric problems in the calculus of variations. I will discuss the basic questions of existence and non-existence of energy minimizers for the isoperimetric problem with a competing non-local term. A complete answer will be given for the case of slowly decaying kernels in two space dimensions, and qualitative properties of the minimizers will be established for general Riesz kernels.