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# Hui, Kin Ming

 My main research area is partial differential equation in particular nonlinear diffusion equations. Nonlinear parabolic equations arise in the modeling of many physical phenomena and in many geometric problems. Recently I mainly work on porous media type equation such as the fast diffusion equation and the logarithmic diffusion equation. Such equation arises in the study of gases or liquid through porous media and in geometric flow problems. I have found that the solutions of such equations have many properties that is different from the solution of the heat equation such as the existence of infinitely many solutions with the same initial value that satisfies the mass loss equation and the extinction of solutions in finite time. I also study global asymptotic behavior and extinction time behavior of the solutions of such equations.   About fifty years ago Feynman delivered a speech entitled, "There's plenty of room at the bottom" in which he described the forthcoming small scale technology including much condensed information storage, miniature computers, infinitesimal machinery, and suggested the possible ways to make them work based on the fundamental principles of physics, chemistry and biology. Micro-electromechanical systems (MEMS) are widely used nowadays including airbag deployment in cars, inkjet printer heads, and the device for the protection of hard disk, etc. The challenge is to understand the mechanism for the various MEMS devices and establish the mathematical models which can reflect and predict their behavior. Recently I also work on MEMS equations and obtain a number of results in this field.        It is well-known that solutions of PDE may has singularities in finite time and it is important to understand the singularities of PDE. 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. 　　 Email : kmhui AT gate.sinica.edu.tw Phone:+886 2 2368-5999 ext. 707 Fax: +886 2 2368-9771

### Publications :

 1 K.M. Hui, A Fatou theorem for the solution of the heat equation at the corner points of a cylinder, Trans. Amer. Math. Soc. , 333(1992), 607-642 2 K.M. Hui, Comparsion theorems for the eigenvalues of the Laplacian in the unit ball in $R^n$, Canad. Math. Bull. , 35(1992), 214-217 3 K.M. Hui, Nonnegative solutions of the fast diffusion equation with strong reaction, Nonlinear analysis TMA , 19(1992), 1155-1178 4 K.M. Hui, Asymptotic behaviour of solutions of $u_t=\Delta u^m-u^p$ as $p\to\infty$, Nonlinear Analysis TMA , 21(1993), 191-195 5 K.M. Hui, Fatou theorem for the solutions of some nonlinear equations, J. Math. Analy. and Appl. , 183(1994), no. 1, 37-52 6 K.M. Hui, Non-existence of fundamental solutions of the equation $u_t=\Delta log u$, J. Math. Anal. and Appl. , 182(1994), 800-809 7 K.M. Hui, Singular limit of solutions of the generalized p-Laplacian equation, Nonlinear Analysis TMA , 24(1995), no. 9, 1327-1345 8 K.M. Hui, Singular limit of solutions $u_t=\Delta u^m-A\cdot\nabla (u^q/q)$ as $q\to\infty$, Trans. Amer. Math. Soc. , 347(1995), no. 5, 1687-1712 9 K.M. Hui, Asymptotic behaviour of solutions of the modified lubrication equation, Applicable Analysis , 62(1996), 311-321 10 K.M.Hui, Singular limit of solutions of some degenerate parabolic equations, Nonlinear waves (Sapporo), GAKUTO Internat. Ser. Math. Sci. Appl. , 10(1995), 161-162 11 K.M. Hui, Singular limit of solutions of the porous medium equation with absorption, Trans. Amer. Math. Soc. , 350(1998), no. 11, 4651-4667 12 K.M. Hui, Singular limit of solutions of the equation $u_t=\Delta(u^m/m)$ as $m\to 0$, Pacific J. Math. , 187(1999), no. 2, 297-316 13 K.M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$, Nonlinear Analysis TMA , 37(1999), 875-914 14 K.M. Hui, Asymptotic behaviour of solutions of $u_t=\Delta\log u$ in a bounded domain, Differential and Integral Equations , 14(2001), no. 2, 175-188 15 K.M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation, Differential and Integral Equations , 15(2002), no. 7, 769-804 16 K.M.Hui, Existence of solutions of the very fast diffusion equation, Nonlinear Analysis TMA , 58(2004), 75-100 17 K.M.Hui, Large time behaviour of solutions of a degenerate fourth order parabolic equation,, Advances in Mathematical Sciences and Applications , 16(2006), no. 2, 467-478 18 K.M. Hui, Existence of solutions of the very fast diffusion equation in bounded and unbounded domain, Mathematische Annalen , 339(2007), no. 2, 395-443 19 K.M. Hui, Singular limit of solutions of the very fast diffusion equation, Nonlinear Analysis, TMA , 68(2008), no. 5, 1120-1147 20 K.M. Hui, Growth rate and extinction rate of a reaction-diffusioin equation with a singular nonlinearity, Differential and Integral Equations , 22(2009), no. 7-8, 771-786 21 K.M. Hui, Global and touchdown behaviour of the generalized MEMS device equation, Advances in Mathematical Sciences and Applications , 19(2009), no. 1, 1-25 22 K.M. Hui, Another proof for the removable singularities of the heat equation, Proceedings Amer. Math. Soc. , 138(2010), no. 7, 2397-2402 23 K.M. Hui, Existence and dynamic properties of a nonlocal MEMS equation, Nonlinear Analysis, TMA , 74(2011), 298-316 24 K.M. Hui, Quenching behaviour of a nonlocal parabolic MEMS equation, Advances in Mathematical Sciences and Applications , 21(2011), no. 1, 173-185 25 (with Sunghoon Kim) Convergence of the Dirichlet solutions of the very fast diffusion equation, Nonlinear Analysis, Theory, Methods and Applications , 74(2011), no. 18, 7404-7425 26 K.M. Hui, Collapsing behaviour of a singular diffusion equation, Discrete and Continuous Dynamical Systems-Series A , 32(2012), no. 6, 2165-2185 27 (with Sunghoon Kim) Large time behaviour of higher dimensional logarithmic diffusion equation, Proceedings of the Royal Society of Edinburgh A , 143(2013), 817-830 28 (with Sunghoon Kim) Extinction profile of the logarithmic diffusion equation, Manuscripta Mathematica , 143(2014), no. 3-4, 491-524 29 (with Sunghoon Kim) Decay rate and radial symmetry of the exponential elliptic equation, J. Math. Anal. Appl. , 413(2014), 269-283 30 (with Sunghoon Kim) Existence of Neumann and singular solutions of the fast diffusion equation, Discrete and Contin. Dynamical Systems-Series A , 35(2015-10), no. 10, 4859-4887 31 (with Sunghoon Kim) Singular limits and properties of solutions of some degenerate elliptic and parabolic equations, Proceedings of the Royal Society of Edinburgh A (to appear) (2017-01) 32 (with Sunghoon Kim) Singular limit of the generalized Burgers equation with absorption (preprint), 12 pages, http://arxiv.org/abs/1501.04253 (2016-12), 0-0 33 (with Soojung Kim) Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete and continuous dynamical systems Series A , 37(2017-11), no. 11, 5943-5977 34 K.M. Hui, Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time, J. Math. Anal. and Appl. , 454(2017-10), no. 2, 695-715 35 K.M. Hui, Global behaviour of solutions of the lubrication equation with porous medium Van der Waals interactions, (in preparation) (preprint), 0-0 36 K.M. Hui, Global behaviour of solution of a nonlinear fourth order parabolic equation, (in preparation) (preprint), 0-0