**Contact Information**

- kmhui$\color{red}{@}$gate.sinica.edu.tw
- +886 2 2368-5999 ext. 707
- +886 2 2368-9771

**Research Interests**

- Partial Differential Equations

**Education**

- Ph.D. University of Chicago (1990)
- M.S. University of Chicago (1986)
- B.S. University of Hong Kong (1984)

**Links**

**Work Experience**

- 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

**Awards**

- 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

**Research Descriptions**

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.

**Selected Publications**

- "K.M. Hui, Global behaviour of solutions of the lubrication equation with porous medium Van der Waals interactions"
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*(in preparation)*, preprint. - "K.M. Hui, Global behaviour of solution of a nonlinear fourth order parabolic equation"
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*(in preparation)*, preprint. - "K.M. Hui, Existence and uniqueness of the singular self-similar solutions of the fast diffusion equation and logarithmic diffusion equation"
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*(preprint)*, 2023-08. (https://arxiv.org/abs/2308.10221) - (with Kai-Seng Chou) "Nonnegative solutions of the heat equation in a cylindrical domain and Widder's theorem"
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*(preprint)*, 2022-10. (https://arxiv.org/abs/2210.10954) - "K.M. Hui, Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions"
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*(preprint)*, 2022-10. (https://arxiv.org/abs/2107.13685) - "K.M. Hui, Existence of hypercylinder expanders of the inverse mean curvature flow"
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*Canadian Math. Bulletin*, 65 (3), 543-551, 2022. - (with Soojung Kim) "Vanishing time behavior of solutions to the fast diffusion equation"
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*(accepted for publication by Communications in Analysis and Geometry)*, 2021-12. (https://arxiv.org/abs/1811.04410) - (with Jinwan Park) "Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured Euclidean space"
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*Discrete and Continuous Dynamical Systems*, 41 (11), 5473-5508, 2021-11. - "K.M. Hui, Uniqueness and time oscillating behaviour of finite points blow-up solutions of the fast diffusion equation"
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*Proceedings of the Royal Society of Edinburgh Section A: Mathematics*, 150 (6), 2849-2870, 2020-12. - (with Sunghoon Kim) "Singular limit of the generalized Burgers equation with absorption"
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*Journal of Dynamics and Differential Equations*, 32, 809-822, 2020. - (with Sunghoon Kim) "Singular limits and properties of solutions of some degenerate elliptic and parabolic equations"
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*Proceedings of the Royal Society of Edinburgh Section A Mathematics*, 149 (2), 353-385, 2019-04. - "K.M. Hui, Existence of self-similar solution of the inverse mean curvature flow"
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*Discrete and Continuous Dynamical Systems--Series A*, 39 (2), 863-890, 2019. - (with Sunghoon Kim) "Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation"
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*Calculus of Variations and PDE, https://doi.org/10.1007/s00526-018-1396-9*, 57 (5), 2018-10. (article 112) - (with Soojung Kim) "Asymptotic large time behavior of singular solutions of the fast diffusion equation"
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*Discrete and continuous dynamical systems Series A*, 37 (11), 5943-5977, 2017-11. - "K.M. Hui, Asymptotic behaviour of solutions of the fast diffusion equation near its extinction time"
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*J. Math. Anal. and Appl.*, 454 (2), 695-715, 2017-10. - (with Sunghoon Kim) "Existence of Neumann and singular solutions of the fast diffusion equation"
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*Discrete and Contin. Dynamical Systems-Series A*, 35 (10), 4859-4887, 2015-10. - (with Sunghoon Kim) "Decay rate and radial symmetry of the exponential elliptic equation"
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*J. Math. Anal. Appl.*, 413, 269-283, 2014. - (with Sunghoon Kim) "Extinction profile of the logarithmic diffusion equation"
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*Manuscripta Mathematica*, 143 (3-4), 491-524, 2014. - (with Sunghoon Kim) "Large time behaviour of higher dimensional logarithmic diffusion equation"
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*Proceedings of the Royal Society of Edinburgh A*, 143, 817-830, 2013. - "K.M. Hui, Collapsing behaviour of a singular diffusion equation"
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*Discrete and Continuous Dynamical Systems-Series A*, 32 (6), 2165-2185, 2012. - (with Sunghoon Kim) "Convergence of the Dirichlet solutions of the very fast diffusion equation"
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*Nonlinear Analysis, Theory, Methods and Applications*, 74 (18), 7404-7425, 2011. - "K.M. Hui, Quenching behaviour of a nonlocal parabolic MEMS equation"
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*Advances in Mathematical Sciences and Applications*, 21 (1), 173-185, 2011. - "K.M. Hui, Existence and dynamic properties of a nonlocal MEMS equation"
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*Nonlinear Analysis, TMA*, 74, 298-316, 2011. - "K.M. Hui, Another proof for the removable singularities of the heat equation"
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*Proceedings Amer. Math. Soc.*, 138 (7), 2397-2402, 2010. - "K.M. Hui, Growth rate and extinction rate of a reaction-diffusioin equation with a singular nonlinearity"
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*Differential and Integral Equations*, 22 (7-8), 771-786, 2009. - "K.M. Hui, Global and touchdown behaviour of the generalized MEMS device equation"
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*Advances in Mathematical Sciences and Applications*, 19 (1), 1-25, 2009. - "K.M. Hui, Singular limit of solutions of the very fast diffusion equation"
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*Nonlinear Analysis, TMA*, 68 (5), 1120-1147, 2008. - "K.M. Hui, Existence of solutions of the very fast diffusion equation in bounded and unbounded domain"
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*Mathematische Annalen*, 339 (2), 395-443, 2007. - "K.M.Hui, Large time behaviour of solutions of a degenerate fourth order parabolic equation,"
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*Advances in Mathematical Sciences and Applications*, 16 (2), 467-478, 2006. - "K.M.Hui, Existence of solutions of the very fast diffusion equation"
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*Nonlinear Analysis TMA*, 58, 75-100, 2004. - "K.M. Hui, On some Dirichlet and Cauchy problems for a singular diffusion equation"
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*Differential and Integral Equations*, 15 (7), 769-804, 2002. - "K.M. Hui, Asymptotic behaviour of solutions of $u_t=\Delta\log u$ in a bounded domain"
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*Differential and Integral Equations*, 14 (2), 175-188, 2001. - "K.M. Hui, Existence of solutions of the equation $u_t=\Delta\log u$"
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*Nonlinear Analysis TMA*, 37, 875-914, 1999. - "K.M. Hui, Singular limit of solutions of the equation $u_t=\Delta(u^m/m)$ as $m\to 0$"
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*Pacific J. Math.*, 187 (2), 297-316, 1999. - "K.M. Hui, Singular limit of solutions of the porous medium equation with absorption"
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*Trans. Amer. Math. Soc.*, 350 (11), 4651-4667, 1998. - "K.M. Hui, Asymptotic behaviour of solutions of the modified lubrication equation"
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*Applicable Analysis*, 62, 311-321, 1996. - "K.M.Hui, Singular limit of solutions of some degenerate parabolic equations"
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*Nonlinear waves (Sapporo), GAKUTO Internat. Ser. Math. Sci. Appl.*, 10, 161-162, 1995. - "K.M. Hui, Singular limit of solutions $u_t=\Delta u^m-A\cdot\nabla (u^q/q)$ as $q\to\infty$"
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*Trans. Amer. Math. Soc.*, 347 (5), 1687-1712, 1995. - "K.M. Hui, Singular limit of solutions of the generalized p-Laplacian equation"
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*Nonlinear Analysis TMA*, 24 (9), 1327-1345, 1995. - "K.M. Hui, Non-existence of fundamental solutions of the equation $u_t=\Delta log u$"
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*J. Math. Anal. and Appl.*, 182, 800-809, 1994. - "K.M. Hui, Fatou theorem for the solutions of some nonlinear equations"
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*J. Math. Analy. and Appl.*, 183 (1), 37-52, 1994. - "K.M. Hui, Asymptotic behaviour of solutions of $u_t=\Delta u^m-u^p$ as $p\to\infty$"
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*Nonlinear Analysis TMA*, 21, 191-195, 1993. - "K.M. Hui, Nonnegative solutions of the fast diffusion equation with strong reaction"
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*Nonlinear analysis TMA*, 19, 1155-1178, 1992. - "K.M. Hui, Comparsion theorems for the eigenvalues of the Laplacian in the unit ball in $R^n$"
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*Canad. Math. Bull.*, 35, 214-217, 1992. - "K.M. Hui, A Fatou theorem for the solution of the heat equation at the corner points of a cylinder"
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*Trans. Amer. Math. Soc.*, 333, 607-642, 1992.