# American Institute of Mathematical Sciences

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 and 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. [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.
 [1] Qi Wang. On some touchdown behaviors of the generalized MEMS device equation. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2447-2456. doi: 10.3934/cpaa.2016043 [2] Nikos I. Kavallaris, Andrew A. Lacey, Christos V. Nikolopoulos, Dimitrios E. Tzanetis. On the quenching behaviour of a semilinear wave equation modelling MEMS technology. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1009-1037. doi: 10.3934/dcds.2015.35.1009 [3] Philip Korman. Infinitely many solutions and Morse index for non-autonomous elliptic problems. Communications on Pure and Applied Analysis, 2020, 19 (1) : 31-46. doi: 10.3934/cpaa.2020003 [4] Zongming Guo, Zhongyuan Liu, Juncheng Wei, Feng Zhou. Bifurcations of some elliptic problems with a singular nonlinearity via Morse index. Communications on Pure and Applied Analysis, 2011, 10 (2) : 507-525. doi: 10.3934/cpaa.2011.10.507 [5] Belgacem Rahal, Cherif Zaidi. On finite Morse index solutions of higher order fractional elliptic equations. Evolution Equations and Control Theory, 2021, 10 (3) : 575-597. doi: 10.3934/eect.2020081 [6] Kelei Wang. Recent progress on stable and finite Morse index solutions of semilinear elliptic equations. Electronic Research Archive, 2021, 29 (6) : 3805-3816. doi: 10.3934/era.2021062 [7] Jiaquan Liu, Yuxia Guo, Pingan Zeng. Relationship of the morse index and the $L^\infty$ bound of solutions for a strongly indefinite differential superlinear system. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 107-119. doi: 10.3934/dcds.2006.16.107 [8] Abdelbaki Selmi, Abdellaziz Harrabi, Cherif Zaidi. Nonexistence results on the space or the half space of $-\Delta u+\lambda u = |u|^{p-1}u$ via the Morse index. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2839-2852. doi: 10.3934/cpaa.2020124 [9] Daniele Cassani, Antonio Tarsia. Periodic solutions to nonlocal MEMS equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 631-642. doi: 10.3934/dcdss.2016017 [10] Ahmet Özkan Özer. Dynamic and electrostatic modeling for a piezoelectric smart composite and related stabilization results. Evolution Equations and Control Theory, 2018, 7 (4) : 639-668. doi: 10.3934/eect.2018031 [11] Laurence Cherfils, Alain Miranville, Shuiran Peng, Chuanju Xu. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1601-1621. doi: 10.3934/dcdss.2019109 [12] Rafael Monteiro. Horizontal patterns from finite speed directional quenching. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3503-3534. doi: 10.3934/dcdsb.2018285 [13] C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88 [14] María Barbero Liñán, Hernán Cendra, Eduardo García Toraño, David Martín de Diego. Morse families and Dirac systems. Journal of Geometric Mechanics, 2019, 11 (4) : 487-510. doi: 10.3934/jgm.2019024 [15] Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006 [16] M. Baake, P. Gohlke, M. Kesseböhmer, T. Schindler. Scaling properties of the Thue–Morse measure. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4157-4185. doi: 10.3934/dcds.2019168 [17] Mauro Patrão, Luiz A. B. San Martin. Morse decomposition of semiflows on fiber bundles. Discrete and Continuous Dynamical Systems, 2007, 17 (3) : 561-587. doi: 10.3934/dcds.2007.17.561 [18] Shu Dai, Dong Li, Kun Zhao. Finite-time quenching of competing species with constrained boundary evaporation. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1275-1290. doi: 10.3934/dcdsb.2013.18.1275 [19] Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control and Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 [20] Yoshikazu Giga, Yukihiro Seki, Noriaki Umeda. On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1463-1470. doi: 10.3934/dcds.2011.29.1463

2020 Impact Factor: 1.392