-
Previous Article
On the integral systems with negative exponents
- DCDS Home
- This Issue
-
Next Article
Attractors and their properties for a class of nonlocal extensible beams
On the quenching behaviour of a semilinear wave equation modelling MEMS technology
1. | Department of Mathematics, School of Science and Engineering, University of Chester, Thornton Science Park, Pool Lane, Ince Chester CH2 4NU, United Kingdom |
2. | Maxwell Institute for Mathematical Sciences & Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, United Kingdom |
3. | Department of Mathematics, University of the Aegean, GR-832 00 Karlovassi, Samos, Greece |
4. | Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece |
References:
[1] |
R. C. Batra, M. Porfiri and D. Spinello, Electromechanical model of electrically actuated narrow microbeams,, Jour. Microelectromechanical Systems, 15 (2006), 1175.
doi: 10.1109/JMEMS.2006.880204. |
[2] |
P. Bizon, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity,, Nonlinearity, 17 (2004), 2187.
doi: 10.1088/0951-7715/17/6/009. |
[3] |
P. Bizon, D. Maison and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity,, Nonlinearity, 20 (2007), 2061.
doi: 10.1088/0951-7715/20/9/003. |
[4] |
C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry,, Journal of Engineering Mathematics, 66 (2010), 217.
doi: 10.1007/s10665-009-9343-6. |
[5] |
P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations,, SIAM Journal of Mathematical Analysis, 12 (1981), 893.
doi: 10.1137/0512075. |
[6] |
C. Y. Chan and K. K. Nip, On the blow-up of $|u_{t t}|$ at quenching for semilinear Euler-Poisson-Darboux equations,, Comp. Appl. Mat., 14 (1995), 185.
|
[7] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010).
|
[8] |
S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quarterly Appl. Math., 51 (1993), 713.
|
[9] |
G. Flores, Dynamics of a damped wave equation arising from MEMS,, SIAM J. Appl. Math., 74 (2014), 1025.
doi: 10.1137/130914759. |
[10] |
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 (2007), 434.
doi: 10.1137/060648866. |
[11] |
V. A. Galaktionov and S. I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation,, Quart. Appl. Math., 61 (2003), 583.
|
[12] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case,, SIAM J. Math. Anal., 38 (2007), 1423.
doi: 10.1137/050647803. |
[13] |
N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS,, Methods Appl. Anal., 15 (2008), 361.
doi: 10.4310/MAA.2008.v15.n3.a8. |
[14] |
J. L. Griffin, S. W. Schlosser, G. R. Ganger and D. F. Nagle, Operating system management of MEMS-based storage devices,, Proceedings of the 4th Symposium on Operating Systems Design and Implementation (OSDI), (2000), 227. Google Scholar |
[15] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behaviour of a MEMS device with varying dielectric properties,, SIAM J. Appl. Math., 66 (2005), 309.
doi: 10.1137/040613391. |
[16] |
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.
|
[17] |
J.-S. Guo and N. I. Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control,, Disc. Cont. Dyn. Systems A, 32 (2012), 1723.
doi: 10.3934/dcds.2012.32.1723. |
[18] |
Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135.
doi: 10.1137/09077117X. |
[19] |
K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation,, Nonl. Anal: Theory, 74 (2011), 298.
doi: 10.1016/j.na.2010.08.045. |
[20] |
N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation,, Nonlinear Diff. Eqns. Appl., 15 (2008), 363.
doi: 10.1007/s00030-008-7081-5. |
[21] |
N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic nonlocal problem modelling MEMS technology,, Rocky Moun. J. Math., 41 (2011), 505.
doi: 10.1216/RMJ-2011-41-2-505. |
[22] |
A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions,, SIAM J. Appl. Math., 43 (1983), 1350.
doi: 10.1137/0143090. |
[23] |
P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS,, Arch. Ration. Mech. Anal., 207 (2013), 139.
doi: 10.1007/s00205-012-0559-7. |
[24] |
P. Laurencot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions,, Proc. London Math. Soc., (). Google Scholar |
[25] |
J. Lega, A. E. Lindsay and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries,, Journal of Nonlinear Science, 23 (2013), 807.
doi: 10.1007/s00332-013-9169-2. |
[26] |
H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations,, Ann. Mat. Pura Appl., 155 (1989), 243.
doi: 10.1007/BF01765943. |
[27] |
H. A. Levine, The phenomenon of quenching: A survey,, Trends in the theory and practice of nonlinear analysis (Arlington, 110 (1985), 275.
doi: 10.1016/S0304-0208(08)72720-8. |
[28] |
H. A. Levine and M. W. Smiley, Abstract wave equations with a singular nonlinear forcing term,, J. Math. Anal. Appl., 103 (1984), 409.
doi: 10.1016/0022-247X(84)90138-0. |
[29] |
C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS,, J. Diff. Equations, 256 (2014), 503.
doi: 10.1016/j.jde.2013.09.010. |
[30] |
C. Liang and K. Zhang, Global solution of the initial boundary value problem to a hyperbolic nonlocal MEMS equation,, Computers & Mathematics with Applications, 67 (2014), 549.
doi: 10.1016/j.camwa.2013.11.012. |
[31] |
A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor,, SIAM J. Appl. Math., 72 (2012), 935.
doi: 10.1137/110832550. |
[32] |
F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension,, J. Funct. Analysis, 253 (2007), 43.
doi: 10.1016/j.jfa.2007.03.007. |
[33] |
J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump,, ASEN 5519 Fluid-Structures Interactions, (5519). Google Scholar |
[34] |
F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations,, Arch. Mat., 45 (2009), 115.
|
[35] |
J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345.
doi: 10.1023/A:1012292311304. |
[36] |
J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties,, SIAM J. Appl. Math., 62 (2002), 888.
doi: 10.1137/S0036139900381079. |
[37] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2003).
|
[38] |
R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions,, SIAM Journal of Math. Analysis, 20 (1989), 1081.
doi: 10.1137/0520072. |
[39] |
H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II,, Theory and Performance, 45 (1994), 67. Google Scholar |
[40] |
J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device,, Patent No.: US 7, (2007). Google Scholar |
[41] |
M. Younis, MEMS Linear and Nonlinear Statics and Dynamics,, Springer, (2011).
doi: 10.1007/978-1-4419-6020-7. |
show all references
References:
[1] |
R. C. Batra, M. Porfiri and D. Spinello, Electromechanical model of electrically actuated narrow microbeams,, Jour. Microelectromechanical Systems, 15 (2006), 1175.
doi: 10.1109/JMEMS.2006.880204. |
[2] |
P. Bizon, T. Chmaj and Z. Tabor, On blowup for semilinear wave equations with a focusing nonlinearity,, Nonlinearity, 17 (2004), 2187.
doi: 10.1088/0951-7715/17/6/009. |
[3] |
P. Bizon, D. Maison and A. Wasserman, Self-similar solutions of semilinear wave equations with a focusing nonlinearity,, Nonlinearity, 20 (2007), 2061.
doi: 10.1088/0951-7715/20/9/003. |
[4] |
C. J. Budd and J. F. Williams, How to adaptively resolve evolutionary singularities in differential equations with symmetry,, Journal of Engineering Mathematics, 66 (2010), 217.
doi: 10.1007/s10665-009-9343-6. |
[5] |
P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations,, SIAM Journal of Mathematical Analysis, 12 (1981), 893.
doi: 10.1137/0512075. |
[6] |
C. Y. Chan and K. K. Nip, On the blow-up of $|u_{t t}|$ at quenching for semilinear Euler-Poisson-Darboux equations,, Comp. Appl. Mat., 14 (1995), 185.
|
[7] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010).
|
[8] |
S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quarterly Appl. Math., 51 (1993), 713.
|
[9] |
G. Flores, Dynamics of a damped wave equation arising from MEMS,, SIAM J. Appl. Math., 74 (2014), 1025.
doi: 10.1137/130914759. |
[10] |
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 (2007), 434.
doi: 10.1137/060648866. |
[11] |
V. A. Galaktionov and S. I. Pohozaev, On similarity solutions and blow-up spectra for a semilinear wave equation,, Quart. Appl. Math., 61 (2003), 583.
|
[12] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case,, SIAM J. Math. Anal., 38 (2007), 1423.
doi: 10.1137/050647803. |
[13] |
N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS,, Methods Appl. Anal., 15 (2008), 361.
doi: 10.4310/MAA.2008.v15.n3.a8. |
[14] |
J. L. Griffin, S. W. Schlosser, G. R. Ganger and D. F. Nagle, Operating system management of MEMS-based storage devices,, Proceedings of the 4th Symposium on Operating Systems Design and Implementation (OSDI), (2000), 227. Google Scholar |
[15] |
Y. Guo, Z. Pan and M. J. Ward, Touchdown and pull-in voltage behaviour of a MEMS device with varying dielectric properties,, SIAM J. Appl. Math., 66 (2005), 309.
doi: 10.1137/040613391. |
[16] |
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.
|
[17] |
J.-S. Guo and N. I. Kavallaris, On a non-local parabolic problem arising in electrostatic MEMS control,, Disc. Cont. Dyn. Systems A, 32 (2012), 1723.
doi: 10.3934/dcds.2012.32.1723. |
[18] |
Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135.
doi: 10.1137/09077117X. |
[19] |
K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation,, Nonl. Anal: Theory, 74 (2011), 298.
doi: 10.1016/j.na.2010.08.045. |
[20] |
N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation,, Nonlinear Diff. Eqns. Appl., 15 (2008), 363.
doi: 10.1007/s00030-008-7081-5. |
[21] |
N. I. Kavallaris, A. A. Lacey, C. V. Nikolopoulos and D. E. Tzanetis, A hyperbolic nonlocal problem modelling MEMS technology,, Rocky Moun. J. Math., 41 (2011), 505.
doi: 10.1216/RMJ-2011-41-2-505. |
[22] |
A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions,, SIAM J. Appl. Math., 43 (1983), 1350.
doi: 10.1137/0143090. |
[23] |
P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS,, Arch. Ration. Mech. Anal., 207 (2013), 139.
doi: 10.1007/s00205-012-0559-7. |
[24] |
P. Laurencot and C. Walker, A fourth-order model for MEMS with clamped boundary conditions,, Proc. London Math. Soc., (). Google Scholar |
[25] |
J. Lega, A. E. Lindsay and F. J. Sayas, The quenching set of a MEMS capacitor in two-dimensional geometries,, Journal of Nonlinear Science, 23 (2013), 807.
doi: 10.1007/s00332-013-9169-2. |
[26] |
H. A. Levine, Quenching, nonquenching, and beyond quenching for solution of some parabolic equations,, Ann. Mat. Pura Appl., 155 (1989), 243.
doi: 10.1007/BF01765943. |
[27] |
H. A. Levine, The phenomenon of quenching: A survey,, Trends in the theory and practice of nonlinear analysis (Arlington, 110 (1985), 275.
doi: 10.1016/S0304-0208(08)72720-8. |
[28] |
H. A. Levine and M. W. Smiley, Abstract wave equations with a singular nonlinear forcing term,, J. Math. Anal. Appl., 103 (1984), 409.
doi: 10.1016/0022-247X(84)90138-0. |
[29] |
C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS,, J. Diff. Equations, 256 (2014), 503.
doi: 10.1016/j.jde.2013.09.010. |
[30] |
C. Liang and K. Zhang, Global solution of the initial boundary value problem to a hyperbolic nonlocal MEMS equation,, Computers & Mathematics with Applications, 67 (2014), 549.
doi: 10.1016/j.camwa.2013.11.012. |
[31] |
A. E. Lindsay and J. Lega, Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor,, SIAM J. Appl. Math., 72 (2012), 935.
doi: 10.1137/110832550. |
[32] |
F. Merle and H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension,, J. Funct. Analysis, 253 (2007), 43.
doi: 10.1016/j.jfa.2007.03.007. |
[33] |
J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump,, ASEN 5519 Fluid-Structures Interactions, (5519). Google Scholar |
[34] |
F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations,, Arch. Mat., 45 (2009), 115.
|
[35] |
J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control,, J. Engrg. Math., 41 (2001), 345.
doi: 10.1023/A:1012292311304. |
[36] |
J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties,, SIAM J. Appl. Math., 62 (2002), 888.
doi: 10.1137/S0036139900381079. |
[37] |
J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2003).
|
[38] |
R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions,, SIAM Journal of Math. Analysis, 20 (1989), 1081.
doi: 10.1137/0520072. |
[39] |
H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II,, Theory and Performance, 45 (1994), 67. Google Scholar |
[40] |
J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device,, Patent No.: US 7, (2007). Google Scholar |
[41] |
M. Younis, MEMS Linear and Nonlinear Statics and Dynamics,, Springer, (2011).
doi: 10.1007/978-1-4419-6020-7. |
[1] |
Bernard Brighi, S. Guesmia. Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain. Conference Publications, 2007, 2007 (Special) : 160-169. doi: 10.3934/proc.2007.2007.160 |
[2] |
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 |
[3] |
Laurence Cherfils, Alain Miranville, Shuiran Peng, Chuanju Xu. Analysis of discretized parabolic problems modeling electrostatic micro-electromechanical systems. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1601-1621. doi: 10.3934/dcdss.2019109 |
[4] |
Patrick Ballard, Bernadette Miara. Formal asymptotic analysis of elastic beams and thin-walled beams: A derivation of the Vlassov equations and their generalization to the anisotropic heterogeneous case. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1547-1588. doi: 10.3934/dcdss.2019107 |
[5] |
S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577 |
[6] |
Jingyu Li, Chuangchuang Liang. Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 833-849. doi: 10.3934/dcds.2016.36.833 |
[7] |
Jong-Shenq Guo, Bo-Chih Huang. Hyperbolic quenching problem with damping in the micro-electro mechanical system device. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 419-434. doi: 10.3934/dcdsb.2014.19.419 |
[8] |
Sigurd Angenent. Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow. Networks & Heterogeneous Media, 2013, 8 (1) : 1-8. doi: 10.3934/nhm.2013.8.1 |
[9] |
Jesús Ildefonso Díaz. On the free boundary for quenching type parabolic problems via local energy methods. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1799-1814. doi: 10.3934/cpaa.2014.13.1799 |
[10] |
Ping Lin, Weihan Wang. Optimal control problems for some ordinary differential equations with behavior of blowup or quenching. Mathematical Control & Related Fields, 2018, 8 (3&4) : 809-828. doi: 10.3934/mcrf.2018036 |
[11] |
Yong Xia. Convex hull of the orthogonal similarity set with applications in quadratic assignment problems. Journal of Industrial & Management Optimization, 2013, 9 (3) : 689-701. doi: 10.3934/jimo.2013.9.689 |
[12] |
M. Chipot, A. Rougirel. On the asymptotic behaviour of the solution of parabolic problems in cylindrical domains of large size in some directions. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 319-338. doi: 10.3934/dcdsb.2001.1.319 |
[13] |
Joachim Escher, Christina Lienstromberg. A survey on second order free boundary value problems modelling MEMS with general permittivity profile. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 745-771. doi: 10.3934/dcdss.2017038 |
[14] |
Hongjing Pan, Ruixiang Xing. On the existence of positive solutions for some nonlinear boundary value problems and applications to MEMS models. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3627-3682. doi: 10.3934/dcds.2015.35.3627 |
[15] |
Matthieu Alfaro, Hiroshi Matano. On the validity of formal asymptotic expansions in Allen-Cahn equation and FitzHugh-Nagumo system with generic initial data. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1639-1649. doi: 10.3934/dcdsb.2012.17.1639 |
[16] |
Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis, Kostas Kouramas. Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers. Numerical Algebra, Control & Optimization, 2019, 0 (0) : 0-0. doi: 10.3934/naco.2019047 |
[17] |
Güher Çamliyurt, Igor Kukavica. A local asymptotic expansion for a solution of the Stokes system. Evolution Equations & Control Theory, 2016, 5 (4) : 647-659. doi: 10.3934/eect.2016023 |
[18] |
Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 |
[19] |
Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 |
[20] |
Moncef Aouadi, Taoufik Moulahi. Asymptotic analysis of a nonsimple thermoelastic rod. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1475-1492. doi: 10.3934/dcdss.2016059 |
2018 Impact Factor: 1.143
Tools
Metrics
Other articles
by authors
[Back to Top]