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On a parabolic equation in MEMS with fringing field

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Abstract

We consider the asymptotic behavior of the solutions to the equation \({u_{t}-u_{xx} = \lambda(1 + {\delta}u_{x}^{2})(1 - u)^{-2}}\) , which comes from Micro-Electromechanical Systems (MEMS) devices modeling. It is shown that when the fringing field exists (i.e., δ > 0), there is a critical value λ * δ > 0 such that if 0 < λ < λ * δ , the equation has a global solution for some initial data; while for λ > λ * δ , all solutions to the equation will quench at finite time. When the quenching happens, u has only finitely many quenching points for particular initial data. A one-side estimate is deduced for the quenching rate of u.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Zhe Liu

  2. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Xiaoliu Wang

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  1. Zhe Liu
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  2. Xiaoliu Wang
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Correspondence to Xiaoliu Wang.

Additional information

This work was supported by PRC Grants NSFC 11101078 and 11171064 and the Natural Science Foundation of Jiangsu Province BK2011583.

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Liu, Z., Wang, X. On a parabolic equation in MEMS with fringing field. Arch. Math. 98, 373–381 (2012). https://doi.org/10.1007/s00013-012-0363-5

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  • DOI: https://doi.org/10.1007/s00013-012-0363-5

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