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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 |
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 (): 1423.
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 (): 434.
doi: 10.1137/060648866. |
[15] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241.
|
[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. |
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. |
[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 (): 1423.
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 (): 434.
doi: 10.1137/060648866. |
[15] |
D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources,, Arch. Rational Mech. Anal., 49 (): 241.
|
[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. |
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