Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the quenching behavior of the MEMS with fringing field


Authors: Xue Luo and Stephen S.-T. Yau
Journal: Quart. Appl. Math. 73 (2015), 629-659
MSC (2000): Primary 35J60, 35B40
DOI: https://doi.org/10.1090/qam/1396
Published electronically: September 10, 2015
MathSciNet review: 3432276
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Abstract: The singular parabolic problem $ u_t-\triangle u=\lambda {\frac {1+\delta \vert\nabla u\vert^2}{(1-u)^2}}$ on a bounded domain $ \Omega $ of $ \mathbb{R}^n$ with Dirichlet boundary condition models the microelectromechanical systems (MEMS) device with fringing field. In this paper, we focus on the quenching behavior of the solution to this equation. We first show that there exists a critical value $ \lambda _\delta ^*>0$ such that if $ 0<\lambda <\lambda _\delta ^*$, all solutions exist globally, while for $ \lambda >\lambda _\delta ^*$, all the solutions will quench in finite time. The estimate of the quenching time in terms of large voltage $ \lambda $ is investigated. Furthermore, the quenching set is a compact subset of $ \Omega $, provided $ \Omega $ is a convex bounded domain in $ \mathbb{R}^n$. In particular, if the domain $ \Omega $ is radially symmetric, then the origin is the only quenching point. We not only derive the one-side estimate of the quenching rate, but also further study the refined asymptotic behavior of the finite quenching solution.


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Additional Information

Xue Luo
Affiliation: School of Mathematics and Systems Science, Beihang University, Haidian District, Beijing, 100191, People’s Republic of China
Email: xluo@buaa.edu.cn, luoxue0327@163.com

Stephen S.-T. Yau
Affiliation: Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China
Email: yau@uic.edu

DOI: https://doi.org/10.1090/qam/1396
Received by editor(s): January 16, 2014
Published electronically: September 10, 2015
Additional Notes: The first author would like to thank the support from the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, the Fundamental Research Funds for the Central Universities (YWF-15-SXXY-006, YWF-15-SXXY-002), and the Beijing Natural Science Foundation (1154011). The second author thanks the support from the start-up fund of Tsinghua University. And both authors appreciate the financial support from the National Natural Science Foundation of China (11471184).
Article copyright: © Copyright 2015 Brown University

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