Rocky Mountain Journal of Mathematics

A hyperbolic non-local problem modelling MEMS technology

N.I. Kavallaris, A.A. Lacey, C.V. Nikolopoulos, and D.E. Tzanetis

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Rocky Mountain J. Math., Volume 41, Number 2 (2011), 505-534.

First available in Project Euclid: 2 May 2011

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Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35J60: Nonlinear elliptic equations
Secondary: 74H35: Singularities, blowup, stress concentrations 74G55: Qualitative behavior of solutions 74K15: Membranes

Electrostatic MEMS quenching of solution hyperbolic non-local problems


Kavallaris, N.I.; Lacey, A.A.; Nikolopoulos, C.V.; Tzanetis, D.E. A hyperbolic non-local problem modelling MEMS technology. Rocky Mountain J. Math. 41 (2011), no. 2, 505--534. doi:10.1216/RMJ-2011-41-2-505.

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  • J.W. Bebernes and A.A. Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Diff. Eqns. 2 (1997), 927-953.
  • –––, Shear band formation for a non-local model of thermo-viscoelastic flows, Adv. Math. Sci. Appl. 15 (2005), 265-282.
  • P. Bizon, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity, Nonlinearity 17 (2004), 2187-2201.
  • P. Bizon, D. Maison and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity, Nonlinearity 20 (2007), 2061-2074.
  • C.Y. Chan and K.K. Nip, On the blow-up of $|u_tt|$ at quenching for semilinear Euler-Poisson-Darboux equations, Comp. Appl. Mat. 14 (1995), 185-190.
  • P.H. Chang and H.A. Levine, The quenching of solutions of semilinear hyperbolic equations, SIAM J. Math. Anal. 12 (1981), 893-903.
  • P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal. 15 (2008), 341-354.
  • P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the first branch of unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), 1731-1768.
  • G. Flores, G. Mercado, J.A. Pelesko and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math. 67 (2007), 434-446.
  • V.A. Galaktionov and S.I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation, Quart. Appl. Math. 61 (2003), 583-600.
  • P.R. Garabedian, Partial differential equations, John Wiley, New York, 1964.
  • N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2007), 1423-1449.
  • –––, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, Nonlinear Diff. Eqns. Appl. 15 (2008), 115-145.
  • F.K. N'Gohisse and Th.K. Boni, Quenching time of some nonlinear wave equations, Arch. Mat. 45 (2009), 115-124.
  • Y. Guo, On the partial differential equations of electrostatic MEMS devices III: Refined touchdown behavior, J. Diff. Equations 244 (2008), 2277-2309.
  • –––, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Diff. Equations 245 (2008), 809-844.
  • J.-S. Guo, B. Hu and C.-J. Wang, A non-local quenching problem arising in micro-electro mechanical systems, Quart. Appl. Math. 67 (2009), 725-734.
  • J.-S. Guo and N.I. Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control, preprint.
  • Y. Guo, Z. Pan and M.J. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math 166 (2006), 309-338.
  • K.M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, Nonlinear Anal. TMA 74 (2011), 298-316.
  • N.I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Diff. Eqns. Appl. 15 (2008), 363-385.
  • H.A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. 155 (1989), 243-260.
  • H.A. Levine and M.W. Smiley, Abstract wave equations with a singular nonlinear forcing term, J. Math. Anal. Appl. 103 (1984), 409-427.
  • J.A. Pelesko, Mathematical modeling of electrostatic MEMS with Taylored dielectric properties, SIAM J. Appl. Math. 62 (2002), 888-908.
  • J.A. Pelesko and D.H. Bernstein, Modeling MEMS and NEMS, Chapman Hall and CRC Press, Boca Raton, 2002.
  • J.A. Pelesko and A.A. Triolo, Non-local problems in MEMS device control, J. Engineering Math. 41 (2001), 345-366.
  • R.A. Smith, On a hyperbolic quenching problem in several dimensions, SIAM J. Math. Anal. 20 (1989), 1081-1094.