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

On a nonlocal parabolic problem arising in electrostatic MEMS control

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.
Citation: Jong-Shenq Guo, Nikos I. Kavallaris. On a nonlocal parabolic problem arising in electrostatic MEMS control. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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. Google Scholar

[8]

J.-S. Guo, Quenching problem in nonhomogeneous media,, Differential and Integral Equations, 10 (1997), 1065. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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 (2004), 221. 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. 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. Google Scholar

[26]

J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345. doi: 10.1023/A:1012292311304. Google Scholar

[27]

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

[29]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007). Google Scholar

[30]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487. 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, 3 (1973), 161. 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. 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. 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. 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. 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. 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. 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. Google Scholar

[8]

J.-S. Guo, Quenching problem in nonhomogeneous media,, Differential and Integral Equations, 10 (1997), 1065. 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. 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. 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. 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. 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. 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, (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. 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. 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. 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. 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. 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. 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 (2004), 221. 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. 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. Google Scholar

[26]

J. A. Pelesko and A. A. Triolo, Nonlocal problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345. doi: 10.1023/A:1012292311304. Google Scholar

[27]

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

[29]

P. Quittner and Ph. Souplet, "Superlinear Parabolic Problems. Blow-Up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007). Google Scholar

[30]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Func. Anal., 7 (1971), 487. 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, 3 (1973), 161. 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. Google Scholar

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