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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), 1175-1189.
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), 2187-2201.
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), 2061-2074.
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), 217-236.
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), 893-903.
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), 185-190. |
[7] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS, Courant Lect. Notes Math. 20, AMS/CIMS, New York, 2010. |
[8] |
S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quarterly Appl. Math., 51 (1993), 713-729. |
[9] |
G. Flores, Dynamics of a damped wave equation arising from MEMS, SIAM J. Appl. Math., 74 (2014), 1025-1035.
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), 434-446.
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), 583-600. |
[12] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.
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), 361-376.
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), 227-242. |
[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), 309-338.
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), 725-734. |
[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), 1723-1746.
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), 1135-1163.
doi: 10.1137/09077117X. |
[19] |
K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, Nonl. Anal: Theory, Methods & Applications, 74 (2011), 298-316.
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), 363-385.
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), 505-534.
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), 1350-1366.
doi: 10.1137/0143090. |
[23] |
P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 207 (2013), 139-158.
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., ().
|
[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), 807-834.
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), 243-260.
doi: 10.1007/BF01765943. |
[27] |
H. A. Levine, The phenomenon of quenching: A survey, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984), North-Holland Math. Stud., North-Holland, Amsterdam, 110 (1985), 275-286.
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), 409-427.
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), 503-530.
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), 549-554.
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), 935-958.
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), 43-121.
doi: 10.1016/j.jfa.2007.03.007. |
[33] |
J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump, ASEN 5519 Fluid-Structures Interactions, 2004. |
[34] |
F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations, Arch. Mat., 45 (2009), 115-124. |
[35] |
J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366.
doi: 10.1023/A:1012292311304. |
[36] |
J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties, SIAM J. Appl. Math., 62 (2002), 888-908.
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), 1081-1094.
doi: 10.1137/0520072. |
[39] |
H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II, Theory and Performance, Sens. Actuat. A, 45 (1994), 67-84. |
[40] |
J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device, Patent No.: US 7,286,155 B1, Date of Patent: Oct. 23, 2007. |
[41] |
M. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 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), 1175-1189.
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), 2187-2201.
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), 2061-2074.
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), 217-236.
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), 893-903.
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), 185-190. |
[7] |
P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS, Courant Lect. Notes Math. 20, AMS/CIMS, New York, 2010. |
[8] |
S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations, Quarterly Appl. Math., 51 (1993), 713-729. |
[9] |
G. Flores, Dynamics of a damped wave equation arising from MEMS, SIAM J. Appl. Math., 74 (2014), 1025-1035.
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), 434-446.
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), 583-600. |
[12] |
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: stationary case, SIAM J. Math. Anal., 38 (2007), 1423-1449.
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), 361-376.
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), 227-242. |
[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), 309-338.
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), 725-734. |
[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), 1723-1746.
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), 1135-1163.
doi: 10.1137/09077117X. |
[19] |
K. M. Hui, Existence and dynamic properties of a parabolic non-local MEMS equation, Nonl. Anal: Theory, Methods & Applications, 74 (2011), 298-316.
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), 363-385.
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), 505-534.
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), 1350-1366.
doi: 10.1137/0143090. |
[23] |
P. Laurencot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS, Arch. Ration. Mech. Anal., 207 (2013), 139-158.
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., ().
|
[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), 807-834.
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), 243-260.
doi: 10.1007/BF01765943. |
[27] |
H. A. Levine, The phenomenon of quenching: A survey, Trends in the theory and practice of nonlinear analysis (Arlington, Tex., 1984), North-Holland Math. Stud., North-Holland, Amsterdam, 110 (1985), 275-286.
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), 409-427.
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), 503-530.
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), 549-554.
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), 935-958.
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), 43-121.
doi: 10.1016/j.jfa.2007.03.007. |
[33] |
J. Nabity, Modeling an Electrostatically Actuated Mems Diaphragm Pump, ASEN 5519 Fluid-Structures Interactions, 2004. |
[34] |
F. K. N'Gohisse and Th. K. Boni, Quenching time of some nonlinear wave equations, Arch. Mat., 45 (2009), 115-124. |
[35] |
J. A. Pelesko and A. A. Triolo, Non-local problems in MEMS device control, J. Engrg. Math., 41 (2001), 345-366.
doi: 10.1023/A:1012292311304. |
[36] |
J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties, SIAM J. Appl. Math., 62 (2002), 888-908.
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), 1081-1094.
doi: 10.1137/0520072. |
[39] |
H. A. C. Tilmans and R. Legtenberg, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II, Theory and Performance, Sens. Actuat. A, 45 (1994), 67-84. |
[40] |
J. I. Trisnadi and C. B. Carlisle, Optical Engine Using One-Dimensional MEMS Device, Patent No.: US 7,286,155 B1, Date of Patent: Oct. 23, 2007. |
[41] |
M. Younis, MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York, 2011.
doi: 10.1007/978-1-4419-6020-7. |
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