## 研究員 |   許健明

 聯絡資訊 Email : kmhui$\color{red}{@}$gate.sinica.edu.tw Phone:+886 2 2368-5999 ext. 707 Fax:    +886 2 2368-9771 相關連結 學歷 Ph.D. 美國芝加哥大學 (1990) M.S. 美國芝加哥大學 (1986) B.S. 香港大學 (1984) 研究專長 偏微分方程

• Research Fellow Institute of Mathematics, Academia Sinica 1999 - Present
• Associate Research Fellow Institute of Mathematics, Academia Sinica 1993/10 - 1999/7
• Assistant Research Fellow Institute of Mathematics, Academia Sinica 1990/10 - 1993/9
• Lecturer in Mathematics University of Chicago 1987 - 1989

• Receipent of the travel grant of the Taipei Trade Centre Exchange Scheme to visit the Department of Mathematics of the University of Hong Kong, 1997
• Recipient of Travel Grants for Young Mathematicians for the International Congress of Mathematicians held in Zurich,, 1994

My main research area is partial differential equations in particular nonlinear diffusion equations. Nonlinear parabolic equations arise in the modeling of many physical phenomena and in many geometric problems. On the other hand singularities arise naturally in the study of partial differential equations and geometric flow problems. For examples in the study of the mean curvature flow, Ricci flow and Yamabe flow, finite time singularities usually occurs. By means of a blow-up argument around the singularities, the solutions of the partial differential equations and geometric flow problems usually approach to some self-similar solutions of the related problems. In the study of the large time behavior or vanishing time behavior of partial differential equations and geometric flow problems, under some appropriate conditions on the initial data and a proper rescaling of the solutions, the rescaled solutions will usually converge to some self-similar solutions of the problems. Hence the study of the singularity formations, existence of various self-similar solutions and particular solutions of the equations and the asymptotic behavior of the solutions are very important topics in the study of partial differential equations.

Recently I mainly study the existence and various properties of the singular solutions and the self-similar solutions of the fast diffusion equation. I also study the relation between the self-similar solutions and the asymptotic large time behaviour and vanishing time behavior of the singular solutions of the fast diffusion equation. Such equation arises from the study of Yamabe flow and physics. For the simplest parabolic equation, the heat equation, I have obtained a necessary and sufficient condition for line singularities for its solution. One of my ongoing project is to understand the singularities of other diffusion type equations.