On a nonlocal parabolic problem arising in electrostatic MEMS control

Jong-Shenq Guo; Nikos I. Kavallaris

doi:10.3934/dcds.2012.32.1723

Article Contents

2012, Volume 32, Issue 5: 1723-1746. Doi: 10.3934/dcds.2012.32.1723

This issue Previous Article Expansive and fixed point free homeomorphisms of the plane Next Article Solitary waves in critical Abelian gauge theories

On a nonlocal parabolic problem arising in electrostatic MEMS control

Jong-Shenq Guo^1, and
Nikos I. Kavallaris^2,

1.
Department of Mathematics, Tamkang University, Tamsui, Taipei County 25137
2.
Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Building B TGr-83200 Karlovassi, Samos

Received: November 30, 2010

Revised: May 31, 2011

Published: April 2012

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Abstract

We consider a nonlocal parabolic equation associated with Dirichlet boundary and initial conditions arising in MEMS control. First, we investigate the structure of the associated steady-state problem for a general star-shaped domain. Then we classify radially symmetric stationary solutions and their radial Morse indices. Finally, we study under which circumstances the solution of the time-dependent problem is global-in-time or quenches in finite time.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 35J60; Secondary: 74H35, 74G55, 74K15.

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References

[1]	M. Al-Refai, N.I. Kavallaris and M. Ali Hajji, Monotone iterative sequences for non-local elliptic problems, Euro. Jnl. Applied Mathematics, 22 (2011), 533-552.doi: 10.1017/S0956792511000246.
[2]	P. Esposito, N. Ghoussoub and Y. Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math., 60 (2007), 1731-1768.doi: 10.1002/cpa.20189.
[3]	P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45.doi: 10.1142/S0219199708002697.
[4]	P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-353.
[5]	B. Gidas, W.-M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.doi: 10.1007/BF01221125.
[6]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal., 38 (2006/07), 1423-1449. doi: 10.1137/050647803.
[7]	N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices. II: Dynamic case, NoDEA Nonlinear Diff. Eqns. Appl., 15 (2008), 115-145.
[8]	J.-S. Guo, Quenching problem in nonhomogeneous media, Differential and Integral Equations, 10 (1997), 1065-1074.
[9]	J.-S. Guo, B. Hu and C.-J. Wang, A nonlocal quenching problem arising in micro-electro mechanical systems, Quarterly Appl. Math., 67 (2009), 725-734.
[10]	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., 66 (2005), 309-338.doi: 10.1137/040613391.
[11]	Y. Guo, On the partial differential equations of electrostatic MEMS devices. III: Refined touchdown behavior, J. Diff. Eqns., 244 (2008), 2277-2309.doi: 10.1016/j.jde.2008.02.005.
[12]	Y. Guo, Global solutions of singular parabolic equations arising from electrostatic MEMS, J. Diff. Eqns., 245 (2008), 809-844.doi: 10.1016/j.jde.2008.03.012.
[13]	Z. Guo and J. Wei, Asymptotic Behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Anal., 7 (2008), 765-786.doi: 10.3934/cpaa.2008.7.765.
[14]	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 (2006/07), 434-446. doi: 10.1137/060648866.
[15]	D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal., 49 (1972/73), 241-269.
[16]	T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966.
[17]	N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, NoDEA Nonlinear Diff. Eqns. Appl., 15 (2008), 363-385.
[18]	N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic non-local problem modelling MEMS technology, Rocky Mountain J. Math., 41 (2011), 505-534.doi: 10.1216/RMJ-2011-41-2-505.
[19]	A. A. Lacey, Thermal runaway in a non-local problem modelling Ohmic heating. I: Model derivation and some special cases, Euro. J. Appl. Math., 6 (1995), 127-144.doi: 10.1017/S095679250000173X.
[20]	H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations, Ann. Mat. Pura Appl. (4), 155 (1989), 243-260.doi: 10.1007/BF01765943.
[21]	C.-S. Lin and W.-M. Ni, A counterexample to the nodal domain conjecture and a related semilinear equation, Proc. Amer. Math. Soc., 102 (1988), 271-277.doi: 10.1090/S0002-9939-1988-0920985-9.
[22]	T. Miyasita, Non-local elliptic problem in higher dimension, Osaka J. Math., 44 (2007), 159-172.
[23]	T. Miyasita and T. Suzuki, Non-local Gel'fand problem in higher dimensions, in "Nonlocal Elliptic and Parabolic Problems," Banach Center Publ., 66, Polish Acad. Sci., Warsaw, (2004), 221-235.
[24]	K. Nagasaki and T. Suzuki, Spectral and related properties about the Emden-Fowler equation $ - \Delta u = \lambda e^u$ on circular domains, Math. Ann., 299 (1994), 1-15.doi: 10.1007/BF01459770.
[25]	Y. Naito and T. Suzuki, Radial symmetry of positive solutions for semilinear elliptic equations on the unit ball in $\R^n$, Funkcial Ekvac., 41 (1998), 215-234.
[26]	J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366.doi: 10.1023/A:1012292311304.
[27]	J. A. Pelesko and D. H. Bernstein, "Modeling MEMS and NEMS," Chapman & Hall/CRC, Boca Raton, FL, 2003.
[28]	S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.
[29]	P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States," Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.
[30]	P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Func. Anal., 7 (1971), 487-513.doi: 10.1016/0022-1236(71)90030-9.
[31]	P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain Consortium Symposium on Nonlinear Eigenvalue Problems (Santa Fe, N.M., 1971), Rocky Mountain J. Math., 3 (1973), 161-202.doi: 10.1216/RMJ-1973-3-2-161.
[32]	R. Schaaf, Uniqueness for semilinear elliptic problems: Supercritical growth and domain geometry, Adv. Diff. Equations, 5 (2000), 1201-1220.