Skip to main content
SpringerLink
Log in

Micro-Electro-Mechanical-Systems (MEMS)

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

  • First Online:
Book cover Non-Local Partial Differential Equations for Engineering and Biology

Part of the book series: Mathematics for Industry ((MFI,volume 31))

  • 1033 Accesses

Abstract

In the current chapter we first present the construction of some non-local models describing the operation of an idealized MEMS (Micro-electro-mechanical system) . In particular, the MEMS device is considered to be part of an electrical circuit and using elastic and electric theories two different non-local models are derived: a parabolic and a hyperbolic one. In the first place, the investigation of the structure of the corresponding non-local elliptic steady state problem is undertaken and some estimates of the pull-in voltage are obtained. Next, we focus on the mathematical analysis of the derived evolutionary non-local equations. Notably, the circumstances under which finite-time quenching occurs for both of evolutionary problems are investigated, so then some useful conclusions regarding the possible destruction of the MEMS device or the invalidity of the used models can be derived. Since, maximum principle is not available for both of the inspected non-local models, and thus comparison methods are not applicable, finally energy methods are called forth to investigate their long-time behavior.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Because (1.170)–(1.172) lacks clear monotonicity properties, it is not obvious that \(\lambda ^*_+ = \lambda ^*_-\).

References

  1. Al-Refai, M., Kavallaris, N.I., Hajji, M.A.: Monotone iterative sequences for non-local elliptic problems. Euro. J. Appl. Math. 22(6), 533–552 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezis, H., Cazenave, T.: A nonlinear heat equation with singular initial data. J. Anal. Math. 68, 277–304 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chan, C.Y., Nip, K.K.: On the blow-up of \(|u_{tt}|\) at quenching for semilinear Euler-Poisson-Darboux equations. Comp. Appl. Mat. 14, 185–190 (1995)

    MATH  MathSciNet  Google Scholar 

  4. Chan, E.K., Dutton, R.W.: Effects of capacitors, resistors and residual change on the static and dynamic performance of electrostatically actuated devices. Proc. SPIE 3680, 120–130 (1999)

    Article  Google Scholar 

  5. Chang, P.H., Levine, H.A.: The quenching of solutions of semilinear hyperbolic equations. SIAM J. Math. Analysis 12, 893–903 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  6. Esposito, P.: Compactness of a nonlinear eigenvalue problem with a singular nonlinearity. Comm. Contemp. Math. 10, 17–45 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Esposito, P., Ghoussoub, N.: Uniqueness of solutions for an elliptic equation modeling MEMS. Methods Appl. Anal. 15, 341–354 (2008)

    MATH  MathSciNet  Google Scholar 

  8. Esposito, P., Ghoussoub, N., Guo, Y.: Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Math. 60, 1731–1768 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Esposito, P., Ghoussoub, N., Guo, Y.: Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS. Courant Lecture Notes in Mathematics, vol. 20. Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence (2010)

    Google Scholar 

  10. Filippas, S., Guo, J.S.: Quenching profiles for one-dimensional semilinear heat equations. Quart. Appl. Math. 51, 713–729 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  11. Friedman, A., McLeod, B.: Blow-up of positive solutions of semilinear heat equations. Ind. Univ. Math. J. 34, 425–447 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  12. Garabedian, P.R.: Partial Differential Equations. Wiley, New York (1964)

    MATH  Google Scholar 

  13. Ghoussoub, N., Guo, Y.: On the partial differential equations of electrostatic MEMS devices: stationary case. SIAM J. Math. Anal. 38, 1423–1449 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gidas, B., Ni, W.-M., Niereberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209–243 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Guo, J.-S.: Quenching problem in nonhomogeneous media. Differ. Integral Equ. 10, 1065–1074 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Guo, J.-S.: Recent developments on a nonlocal problem arising in the micro-electro mechanical system. Tamkang J. Math. 45, 229–241 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  17. Guo, J.-S., Kavallaris, N.I.: On a nonlocal parabolic problem arising in electromechanical MEMS control. Disc. Cont. Dyn. Systems 32, 1723–1746 (2012)

    Article  MATH  Google Scholar 

  18. Guo, J.-S., Hu, B., Wang, C.-J.: A nonlocal quenching problem arising in micro-electro mechanical systems. Quart. Appl. Math. 67, 725–734 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Guo, Y., Pan, Z., Ward, M.J.: Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties. SIAM J. Appl. Math. 166, 309–338 (2006)

    MATH  MathSciNet  Google Scholar 

  20. Guo, Z., Wei, J.: Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity. Comm. Pure Appl. Anal. 7, 765–786 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hui, K.-M.: The existence and dynamic properties of a parabolic nonlocal MEMS equation. Nonl. Anal. (TMA) 74, 298–316 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. Joseph, D.D., Lundgren, T.S.: Quasilinear dirichlet problems driven by positive sources. Arch. Ration. Mech. Anal. 49, 241–269 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kaplan, S.: On the growth of solutions of quasilinear parabolic equations. Comm. Pure Appl. Maths. 16, 327–330 (1963)

    Article  Google Scholar 

  24. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

    Book  MATH  Google Scholar 

  25. Kavallaris, N.I.: Quenching solutions of a stochastic parabolic problem arising in electrostatic MEMS control. Math. Meth. Appl. Sci (2016). https://doi.org/10.1002/mma.4176

  26. Kavallaris, N.I., Nadzieja, T.: On the blow-up of the non-local thermistor problem. Proc. Edin. Math. Soc. 50, 389–409 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kavallaris, N.I., Miyasita, T., Suzuki, T.: Touchdown and related problems in electrostatic MEMS device equation. Non. Diff. Eqns. Appl. 15, 363–385 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kavallaris, N.I., Lacey, A.A., Nikolopoulos, C.V., Tzanetis, D.E.: A hyperbolic nonlocal problem modelling MEMS technology. Rocky Mountain J. Math. 41, 505–534 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kavallaris, N.I., Lacey, A.A., Nikolopoulos, C.V.: On the quenching of a nonlocal parabolic problem arising in electrostatic MEMS control. Nonl. Anal. 138, 189–206 (2016). Nonlinear Partial Differential Equations, in honor of Juan Luis Váquez for his 70th birthday

    Article  MATH  MathSciNet  Google Scholar 

  30. Lacey, A.A.: Thermal runaway in a non-local problem modelling Ohmic heating. Part I: model derivation and some specail cases. Euro. J. Appl. Math. 6, 127–144 (1995)

    MATH  Google Scholar 

  31. Levine, H.A.: Quenching, nonquenching, and beyond quenching for solution of some parabolic equations. Ann. Mat. Pura Appl. 155, 243–260 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lin, C.-S., Ni, W.-M.: A counterexample to the nodal domain conjecture and a related semilinear equation. Proc. Amer. Math. Soc. 102, 271–277 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  33. Miyasita, T.: Non-local elliptic problem in higher dimension. Osaka J. Math. 44, 159–172 (2007)

    MATH  MathSciNet  Google Scholar 

  34. Nagasaki, K., Suzuki, T.: Spectral and related properties about the Emden-Fowler equation \( - \Delta u = \lambda e^u\) on circular domains. Math. Ann. 299, 1–15 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Naito, Y., Suzuki, T.: Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in \(\mathbb{R}^n\). Funkcial Ekvac. 41, 215–234 (1998)

    MATH  MathSciNet  Google Scholar 

  36. Pelesko, J.A., Bernstein, D.H.: Modeling MEMS and NEMS. Chapman Hall and CRC Press, Boca Raton (2002)

    Book  MATH  Google Scholar 

  37. Pelesko, J.A., Triolo, A.A.: Nonlocal problems in MEMS device control. J. Eng. Math. 41, 345–366 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pohozaev, S.I.: Eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\). Doklady 165, 1408–1411 (1965)

    MATH  Google Scholar 

  39. Quittner, P., Souplet, Ph: Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States. Birkhäuser Advanced Texts. Birkhäuser Verlag, Basel (2007)

    MATH  Google Scholar 

  40. Rabinowitz, P.H.: Some global results for nonlinear eigenvalue problems. J. Func. Anal. 7, 487–513 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  41. Rabinowitz, P.H.: Some aspects of nonlinear eigenvalue problems. Rocky Mountain J. Math. 3, 161–202 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  42. Seeger, J.J., Crary, S.B.: Stabilization of electrostatically actuated mechanical devices. In: Proceedings of the 1997 International Conference on Solid-State Sensors and Actuators, pp. 1133–1336 (1997)

    Google Scholar 

  43. Seeger, J.J., Crary, S.B.: Analysis and simulation of MOS capacitor feedback for stabilizing electrostatically actuated, echanical devices. In: Second International Conference on the Simulation and Design of Microsystems and Microstructures-MICROSIM97, pp. 199–208 (1997)

    Google Scholar 

  44. Schaaf, R.: Uniqueness for semilinear elliptic problems supercritical growth and domain geometry. Adv. Diff. Equ. 5, 1201–1220 (2000)

    MATH  MathSciNet  Google Scholar 

  45. Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in \(L^p\). Indiana Univ. Math. J. 29, 79–102 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  46. Younis, M.: MEMS Linear and Nonlinear Statics and Dynamics. Springer, New York (2011)

    Book  Google Scholar 

  47. Zheng, G.-F.: Some dichotomy results for the quenching problem. Acta Math. Sinica 28, 1491–1506 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Chester, Chester, UK

    Nikos I. Kavallaris

  2. Department of Mathematics, Osaka University, Osaka, Japan

    Takashi Suzuki

Authors
  1. Nikos I. Kavallaris
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takashi Suzuki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikos I. Kavallaris .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Kavallaris, N.I., Suzuki, T. (2018). Micro-Electro-Mechanical-Systems (MEMS). In: Non-Local Partial Differential Equations for Engineering and Biology. Mathematics for Industry, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-319-67944-0_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-67944-0_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-67942-6

  • Online ISBN: 978-3-319-67944-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation