# American Institute of Mathematical Sciences

March  2015, 35(3): 1009-1037. doi: 10.3934/dcds.2015.35.1009

## 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

Received  March 2014 Revised  July 2014 Published  October 2014

In this work we study the semilinear wave equation of the form $u_{tt}=u_{xx} + {\lambda}/{ (1-u)^2},$ with homogeneous Dirichlet boundary conditions and suitable initial conditions, which, under appropriate circumstances, serves as a model of an idealized electrostatically actuated MEMS device. First we establish local existence of the solutions of the problem for any $\lambda>0.$ Then we focus on the singular behaviour of the solution, which occurs through finite-time quenching, i.e. when $||u(\cdot,t)||_{\infty}\to 1$ as $t\to t^*- < \infty$, investigating both conditions for quenching and the quenching profile of $u.$ To this end, the non-existence of a regular similarity solution near a quenching point is first shown and then a formal asymptotic expansion is used to determine the local form of the solution. Finally, using a finite difference scheme, we solve the problem numerically, illustrating the preceding results.
Citation: 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 & Continuous Dynamical Systems - A, 2015, 35 (3) : 1009-1037. doi: 10.3934/dcds.2015.35.1009
##### 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010). Google Scholar [8] S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quarterly Appl. Math., 51 (1993), 713. Google Scholar [9] G. Flores, Dynamics of a damped wave equation arising from MEMS,, SIAM J. Appl. Math., 74 (2014), 1025. doi: 10.1137/130914759. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135. doi: 10.1137/09077117X. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions,, SIAM J. Appl. Math., 43 (1983), 1350. doi: 10.1137/0143090. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties,, SIAM J. Appl. Math., 62 (2002), 888. doi: 10.1137/S0036139900381079. Google Scholar [37] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2003). Google Scholar [38] R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions,, SIAM Journal of Math. Analysis, 20 (1989), 1081. doi: 10.1137/0520072. Google Scholar [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. Google Scholar

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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010). Google Scholar [8] S. Fillipas and J.-S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quarterly Appl. Math., 51 (1993), 713. Google Scholar [9] G. Flores, Dynamics of a damped wave equation arising from MEMS,, SIAM J. Appl. Math., 74 (2014), 1025. doi: 10.1137/130914759. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [18] Y. Guo, Dynamical solutions of singular wave equations modeling electrostatic MEMS,, SIAM J. Appl. Dyn. Syst., 9 (2010), 1135. doi: 10.1137/09077117X. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] A. A. Lacey, Mathematical analysis of thermal runaway for spatially inhomogeneous reactions,, SIAM J. Appl. Math., 43 (1983), 1350. doi: 10.1137/0143090. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] J. A. Pelesko, Mathematical Modeling of Electrostatic MEMS with Taylored Dielectric Properties,, SIAM J. Appl. Math., 62 (2002), 888. doi: 10.1137/S0036139900381079. Google Scholar [37] J. A. Pelesko and D. H. Bernstein, Modeling MEMS and NEMS,, Chapman Hall and CRC Press, (2003). Google Scholar [38] R. A. Smith, On A Hyperbolic Quenching Problem In Several Dimensions,, SIAM Journal of Math. Analysis, 20 (1989), 1081. doi: 10.1137/0520072. Google Scholar [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. Google Scholar
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