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
Full-text PDF Free Access
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Abstract: The singular parabolic problem $u_t-\triangle u=\lambda {\frac {1+\delta |\nabla u|^2}{(1-u)^2}}$ 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.
References
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References
- Haïm Brezis, Thierry Cazenave, Yvan Martel, and Arthur Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations 1 (1996), no. 1, 73–90. MR 1357955 (96i:35063)
- Marek Fila and Josephus Hulshof, A note on the quenching rate, Proc. Amer. Math. Soc. 112 (1991), no. 2, 473–477. MR 1055772 (92a:35090), DOI https://doi.org/10.2307/2048742
- Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964. MR 0181836 (31 \#6062)
- Avner Friedman and Bryce McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985), no. 2, 425–447. MR 783924 (86j:35089), DOI https://doi.org/10.1512/iumj.1985.34.34025
- Nassif Ghoussoub and Yujin Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1423–1449 (electronic). MR 2286013 (2007m:35063), DOI https://doi.org/10.1137/050647803
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- Nassif Ghoussoub and Yujin Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS, Methods Appl. Anal. 15 (2008), no. 3, 361–376. MR 2500853 (2010g:35184), DOI https://doi.org/10.4310/MAA.2008.v15.n3.a8
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879 (80h:35043)
- Yoshikazu Giga and Robert V. Kohn, Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math. 38 (1985), no. 3, 297–319. MR 784476 (86k:35065), DOI https://doi.org/10.1002/cpa.3160380304
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- Yujin Guo, Zhenguo Pan, and M. J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 66 (2005), no. 1, 309–338 (electronic). MR 2179754 (2006f:35130), DOI https://doi.org/10.1137/040613391
- Joseph B. Keller and John S. Lowengrub, Asymptotic and numerical results for blowing-up solutions to semilinear heat equations, Singularities in fluids, plasmas and optics (Heraklion, 1992) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 404, Kluwer Acad. Publ., Dordrecht, 1993, pp. 111–129. MR 1864363
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- Wen Xiong Liu, The blow-up rate of solutions of semilinear heat equations, J. Differential Equations 77 (1989), no. 1, 104–122. MR 980545 (90e:35022), DOI https://doi.org/10.1016/0022-0396%2889%2990159-9
- Zhe Liu and Xiaoliu Wang, On a parabolic equation in MEMS with fringing field, Arch. Math. (Basel) 98 (2012), no. 4, 373–381. MR 2914353, DOI https://doi.org/10.1007/s00013-012-0363-5
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- John A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math. 62 (2001/02), no. 3, 888–908 (electronic). MR 1897727 (2003b:74025), DOI https://doi.org/10.1137/S0036139900381079
- John A. Pelesko and David H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, FL, 2003. MR 1955412 (2003m:74004)
- John A. Pelesko and Tobin A. Driscoll, The effect of the small-aspect-ratio approximation on canonical electrostatic MEMS models, J. Engrg. Math. 53 (2005), no. 3-4, 239–252. MR 2230109 (2006m:74113), DOI https://doi.org/10.1007/s10665-005-9013-2
- L. Shampine, J. Kierzenka, and M. Reichelt, Solving boundary value problems for ordinary differential equations in MATLAB with bvp4c, available at http://www.mathworks.com/bvp_{t}utorial.
- Qi Wang, Estimates for the quenching time of a MEMS equation with fringing field, J. Math. Anal. Appl. 405 (2013), no. 1, 135–147. MR 3053493, DOI https://doi.org/10.1016/j.jmaa.2013.03.054
- Juncheng Wei and Dong Ye, On MEMS equation with fringing field, Proc. Amer. Math. Soc. 138 (2010), no. 5, 1693–1699. MR 2587454 (2011d:35072), DOI https://doi.org/10.1090/S0002-9939-09-10226-5
- Dong Ye and Feng Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differential Equations 37 (2010), no. 1-2, 259–274. MR 2564407 (2011e:35184), DOI https://doi.org/10.1007/s00526-009-0262-1
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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
MR Author ID:
185485
Email:
yau@uic.edu
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).
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Brown University