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May  2012, 32(5): 1723-1746. doi: 10.3934/dcds.2012.32.1723

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, Taiwan
2. 

Department of Statistics and Actuarial-Financial Mathematics, University of the Aegean, Building B TGr-83200 Karlovassi, Samos, Greece

Received  December 2010 Revised  June 2011 Published  January 2012

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: Electrostatic MEMS, quenching, touchdown, Morse index..
Mathematics Subject Classification: Primary: 35K55, 35J60; Secondary: 74H35, 74G55, 74K1.
Citation: Jong-Shenq Guo, Nikos I. Kavallaris. On a nonlocal parabolic problem arising in electrostatic MEMS control. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1723-1746. doi: 10.3934/dcds.2012.32.1723
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.  Google Scholar

[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.  Google Scholar

[3]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697.  Google Scholar

[4]

P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-353.  Google Scholar

[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.  Google Scholar

[6]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case,, SIAM J. Math. Anal., 38 (): 1423.  doi: 10.1137/050647803.  Google Scholar

[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.  Google Scholar

[8]

J.-S. Guo, Quenching problem in nonhomogeneous media, Differential and Integral Equations, 10 (1997), 1065-1074.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 (): 434.  doi: 10.1137/060648866.  Google Scholar

[15]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241.   Google Scholar

[16]

T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. A. Pelesko and D. H. Bernstein, "Modeling MEMS and NEMS," Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[28]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Schaaf, Uniqueness for semilinear elliptic problems: Supercritical growth and domain geometry, Adv. Diff. Equations, 5 (2000), 1201-1220.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[3]

P. Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Comm. Contemp. Math., 10 (2008), 17-45. doi: 10.1142/S0219199708002697.  Google Scholar

[4]

P. Esposito and N. Ghoussoub, Uniqueness of solutions for an elliptic equation modeling MEMS, Methods Appl. Anal., 15 (2008), 341-353.  Google Scholar

[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.  Google Scholar

[6]

N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case,, SIAM J. Math. Anal., 38 (): 1423.  doi: 10.1137/050647803.  Google Scholar

[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.  Google Scholar

[8]

J.-S. Guo, Quenching problem in nonhomogeneous media, Differential and Integral Equations, 10 (1997), 1065-1074.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 (): 434.  doi: 10.1137/060648866.  Google Scholar

[15]

D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241.   Google Scholar

[16]

T. Kato, "Perturbation Theory for Linear Operators," Springer, Berlin, 1966. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

J. A. Pelesko and D. H. Bernstein, "Modeling MEMS and NEMS," Chapman & Hall/CRC, Boca Raton, FL, 2003.  Google Scholar

[28]

S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

R. Schaaf, Uniqueness for semilinear elliptic problems: Supercritical growth and domain geometry, Adv. Diff. Equations, 5 (2000), 1201-1220.  Google Scholar

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