Chinese Annals of Mathematics, Series B

, Volume 39, Issue 1, pp 129–144 | Cite as

Quenching phenomenon for a parabolic MEMS equation

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Abstract

This paper deals with the electrostatic MEMS-device parabolic equation
$${u_t} - \Delta u = \frac{{\lambda f(x)}}{{{{(1 - u)}^p}}}$$
in a bounded domain Ω of ℝ N , with Dirichlet boundary condition, an initial condition u0(x) ∈ [0, 1) and a nonnegative profile f, where λ > 0, p > 1. The study is motivated by a simplified micro-electromechanical system (MEMS for short) device model. In this paper, the author first gives an asymptotic behavior of the quenching time T* for the solution u to the parabolic problem with zero initial data. Secondly, the author investigates when the solution u will quench, with general λ, u0(x). Finally, a global existence in the MEMS modeling is shown.

Keywords

MEMS equation Quenching time Global existence 

2000 MR Subject Classification

35A01 35B44 35K58 
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References

  1. [1]
    Ambrosetti, A. and Prodi, G., A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, Massachusetts, 1995.MATHGoogle Scholar
  2. [2]
    Brauner, C. M. and Nicolaenko, B., Sur une classe de problemes elliptiques non lineaires, C. R. Acad. Sci. Paris A, 286, 1978, 1007–1010.MathSciNetMATHGoogle Scholar
  3. [3]
    Brezis, H., Cazenave, T., Martel, Y. and Ramiandrisoa, A., Blow-up for ut-≦u = g(u) revisited, Advances in Differential Equations, 1, 1996, 73–90.MathSciNetMATHGoogle Scholar
  4. [4]
    Cantrell, R. S. and Cosner, C., Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003.MATHGoogle Scholar
  5. [5]
    Esposito, P. and Ghoussoub, N., Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15(3), 2008, 341–354.MathSciNetMATHGoogle Scholar
  6. [6]
    Esposito, P., Ghoussoub, N. and Guo, Y. J., Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60(12), 2007, 1731–1768.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    Esposito, P., Ghoussoub, N. and Guo, Y. J., Mathematical analysis of partial differential equations mod- eling electrostatic MEMS, Courant Lecture Notes in Mathematics, 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010.Google Scholar
  8. [8]
    Flores, G., Mercado, G. A. and Pelesko, J. A., Dynamics and touchdown in electrostatic MEMS, Proceed- ings of ASME DETC’03, IEEE Computer Soc., 2003, 1–8.Google Scholar
  9. [9]
    Ghoussoub, N. and Guo, Y. J., Estimates for the quenching time of a parabolic equation modeling elec- trostatic MEMS, Methods Appl. Anal., 15(3), 2008, 361–376.MathSciNetMATHGoogle Scholar
  10. [10]
    Ghoussoub, N. and Guo, Y. J., On the partial differential equations of electrostatic MEMS devices: Sta- tionary case, SIAM J. Math. Anal., 38, 2007, 1423–1449.CrossRefMATHGoogle Scholar
  11. [11]
    Ghoussoub, N. and Guo, Y. J., On the partial differential equations of electrostatic MEMS devices II: Dynamic case, NoDEA Nonlinear Differ. Equ. Appl., 15, 2008, 115–145.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Ghoussoub, N. and Guo, Y. J., On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior, J. Differ. Equ., 244, 2008, 2277–2309.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Guo, Y. J., Pan, Z. G. and Ward, M. J., Touchdown and pull-in voltage behavior of an MEMS device with varying dielectric properties, SIAM J. Appl. Math., 66(1), 2005, 309–338.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    Hu, B., Blow-up Theories for Semilinear Parabolic Equations, Springer-Verlag, Berlin, 2011.CrossRefMATHGoogle Scholar
  15. [15]
    Lacey, A. A., Mathematical analysis of thermal runaway for spatially inhomogeneous reactions, SIAM J. Appl. Math., 43, 1983, 1350–1366.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    Lin, F. H. and Yang, Y. S., Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A, 463, 2007, 1323–1337.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Mignot, F. and Puel, J. P., Sur une classe de probleme non lineaires avec non linearite positive, croissante, convexe, Comm. P.D.E., 5(8), 1980, 791–836.CrossRefMATHGoogle Scholar
  18. [18]
    Pelesko, J. A. and Berstein, D. H., Modeling MEMS and NEMS, Chapman & Hall/CRC, Boca Raton, 2002.CrossRefGoogle Scholar
  19. [19]
    Pelesko, J. A., Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62(3), 2002, 888–908.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    Sattinger, D. H., Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21, 1972, 979–1000.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    Wang, Q., Estimates for the quenching time of an MEMS equation with fringing field, J. Math. Anal. Appl., 405, 2013, 135–147.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    Ye, D. and Zhou, F., On a general family of nonautonomous elliptic and parabolic equations, Calc. Var., 37, 2010, 259–274.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of ScienceUniversity of Shanghai for Science and TechnologyShanghaiChina

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