# American Institute of Mathematical Sciences

February  2016, 36(2): 833-849. doi: 10.3934/dcds.2016.36.833

## Viscosity dominated limit of global solutions to a hyperbolic equation in MEMS

 1 School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China 2 Department of Mathematics, Capital Normal University, Beijing 100037, China

Received  July 2014 Revised  January 2015 Published  August 2015

We study the asymptotic relation of solutions between the hyperbolic equation and the parabolic one over a one-dimensional bounded interval, both of which model a simple electrostatic micro-electro-mechanical system (MEMS) device. The relation is characterized by a limit as a physical parameter representing the strength of inertial forces tends to zero. We call this limit the viscosity dominated limit. It is shown that in this singular limit the solution of the hyperbolic model converges to that of the parabolic one globally in time. Also the higher order terms including the initial layer corrections, as well as the related error estimates, are derived. Furthermore, it is proved that the convergence is valid for global solutions with large initial data.
Citation: 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
##### References:
 [1] P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations,, SIAM J. Math. Anal., 12 (1981), 893. doi: 10.1137/0512075. Google Scholar [2] J. Escher, P. Laurençot and C. Walker, A parabolic free boundary problem modeling electrostatic MEMS,, Arch. Rat. Mech. Anal., 211 (2014), 389. doi: 10.1007/s00205-013-0656-2. Google Scholar [3] 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. doi: 10.1002/cpa.20189. Google Scholar [4] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010). Google Scholar [5] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar [6] S. Filippas and J. S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quart. Appl. Math., 51 (1993), 713. Google Scholar [7] G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and Touchdown in Electrostatic MEMS,, Proceedings of ICMENS, (2003), 162. Google Scholar [8] 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. Google Scholar [9] N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS,, Methods Appl. Anal., 15 (2008), 361. Google Scholar [10] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices, II. Dynamic case,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 115. doi: 10.1007/s00030-007-6004-1. Google Scholar [11] J. S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation,, J. Math. Anal. Appl., 151 (1990), 58. doi: 10.1016/0022-247X(90)90243-9. Google Scholar [12] J. S. Guo, On the quenching rate estimate,, Quart. Appl. Math., 49 (1991), 747. Google Scholar [13] J. S. Guo, On a quenching problem with the Robin boundary condition,, Nonlinear Anal., 17 (1991), 803. doi: 10.1016/0362-546X(91)90154-S. Google Scholar [14] J. S. Guo, Quenching problem in nonhomogeneous media,, Differential Integral Equations, 10 (1997), 1065. Google Scholar [15] J. S. Guo, B. Hu and C. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system,, Quart. Appl. Math., 67 (2009), 725. doi: 10.1090/S0033-569X-09-01159-5. Google Scholar [16] J. S. Guo and B. Huang, Hyperbolic quenching problem with damping in the micro-electro mechanical system device,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 419. doi: 10.3934/dcdsb.2014.19.419. Google Scholar [17] J. S. Guo and N. Kavallaris, On a nonlocal parabolic problem arising in electrostatic MEMS control,, Discrete Contin. Dyn. Syst., 32 (2012), 1723. doi: 10.3934/dcds.2012.32.1723. Google Scholar [18] Y. Guo, On the partial differential equations of electrostatic MEMS devices, III. Refined touchdown behavior,, J. Differential Equations, 244 (2008), 2277. doi: 10.1016/j.jde.2008.02.005. Google Scholar [19] 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 [20] Z. M. Guo and J. Wei, Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 765. doi: 10.3934/cpaa.2008.7.765. Google Scholar [21] Z. M. Guo and J. Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity,, J. Lond. Math. Soc., 78 (2008), 21. doi: 10.1112/jlms/jdm121. Google Scholar [22] S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: 10.1002/cpa.3160160307. Google Scholar [23] 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. doi: 10.1216/RMJ-2011-41-2-505. Google Scholar [24] H. Kawarada, On solutions of initial boundary value problem for $u_t=u_{x x}+\frac{1}{1-u}$,, RIMS Kyoto U., 10 (1975), 729. doi: 10.2977/prims/1195191889. Google Scholar [25] P. Laurençot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS,, Arch. Rat. Mech. Anal., 207 (2013), 139. doi: 10.1007/s00205-012-0559-7. 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] C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS,, J. Differential Equations, 256 (2014), 503. doi: 10.1016/j.jde.2013.09.010. Google Scholar [28] C. Liang and K. Zhang, Asymptotic stability and quenching behavior of a hyperbolic nonlocal MEMS equation,, Commun. Math. Sci., 13 (2015), 355. doi: 10.4310/CMS.2015.v13.n2.a5. Google Scholar [29] J. A. Pelesko and A. A. Bernstein, Modeling MEMS and NEMS,, Chapman and Hall, (2003). Google Scholar [30] R. A. Smith, On a hyperbolic quenching problem in several dimensions,, SIAM J. Math. Anal., 20 (1989), 1081. doi: 10.1137/0520072. Google Scholar [31] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 37 (2010), 259. doi: 10.1007/s00526-009-0262-1. Google Scholar

show all references

##### References:
 [1] P. H. Chang and H. A. Levine, The quenching of solutions of semilinear hyperbolic equations,, SIAM J. Math. Anal., 12 (1981), 893. doi: 10.1137/0512075. Google Scholar [2] J. Escher, P. Laurençot and C. Walker, A parabolic free boundary problem modeling electrostatic MEMS,, Arch. Rat. Mech. Anal., 211 (2014), 389. doi: 10.1007/s00205-013-0656-2. Google Scholar [3] 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. doi: 10.1002/cpa.20189. Google Scholar [4] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical Analysis of Partial Differential Equations Modeling Electrostatic MEMS,, Courant Lect. Notes Math. 20, (2010). Google Scholar [5] L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998). Google Scholar [6] S. Filippas and J. S. Guo, Quenching profiles for one-dimensional semilinear heat equations,, Quart. Appl. Math., 51 (1993), 713. Google Scholar [7] G. Flores, G. A. Mercado and J. A. Pelesko, Dynamics and Touchdown in Electrostatic MEMS,, Proceedings of ICMENS, (2003), 162. Google Scholar [8] 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. Google Scholar [9] N. Ghoussoub and Y. Guo, Estimates for the quenching time of a parabolic equation modeling electrostatic MEMS,, Methods Appl. Anal., 15 (2008), 361. Google Scholar [10] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices, II. Dynamic case,, NoDEA Nonlinear Differential Equations Appl., 15 (2008), 115. doi: 10.1007/s00030-007-6004-1. Google Scholar [11] J. S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation,, J. Math. Anal. Appl., 151 (1990), 58. doi: 10.1016/0022-247X(90)90243-9. Google Scholar [12] J. S. Guo, On the quenching rate estimate,, Quart. Appl. Math., 49 (1991), 747. Google Scholar [13] J. S. Guo, On a quenching problem with the Robin boundary condition,, Nonlinear Anal., 17 (1991), 803. doi: 10.1016/0362-546X(91)90154-S. Google Scholar [14] J. S. Guo, Quenching problem in nonhomogeneous media,, Differential Integral Equations, 10 (1997), 1065. Google Scholar [15] J. S. Guo, B. Hu and C. Wang, A nonlocal quenching problem arising in a micro-electro mechanical system,, Quart. Appl. Math., 67 (2009), 725. doi: 10.1090/S0033-569X-09-01159-5. Google Scholar [16] J. S. Guo and B. Huang, Hyperbolic quenching problem with damping in the micro-electro mechanical system device,, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 419. doi: 10.3934/dcdsb.2014.19.419. Google Scholar [17] J. S. Guo and N. Kavallaris, On a nonlocal parabolic problem arising in electrostatic MEMS control,, Discrete Contin. Dyn. Syst., 32 (2012), 1723. doi: 10.3934/dcds.2012.32.1723. Google Scholar [18] Y. Guo, On the partial differential equations of electrostatic MEMS devices, III. Refined touchdown behavior,, J. Differential Equations, 244 (2008), 2277. doi: 10.1016/j.jde.2008.02.005. Google Scholar [19] 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 [20] Z. M. Guo and J. Wei, Asymptotic behavior of touch-down solutions and global bifurcations for an elliptic problem with a singular nonlinearity,, Commun. Pure Appl. Anal., 7 (2008), 765. doi: 10.3934/cpaa.2008.7.765. Google Scholar [21] Z. M. Guo and J. Wei, Infinitely many turning points for an elliptic problem with a singular non-linearity,, J. Lond. Math. Soc., 78 (2008), 21. doi: 10.1112/jlms/jdm121. Google Scholar [22] S. Kaplan, On the growth of solutions of quasi-linear parabolic equations,, Comm. Pure Appl. Math., 16 (1963), 305. doi: 10.1002/cpa.3160160307. Google Scholar [23] 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. doi: 10.1216/RMJ-2011-41-2-505. Google Scholar [24] H. Kawarada, On solutions of initial boundary value problem for $u_t=u_{x x}+\frac{1}{1-u}$,, RIMS Kyoto U., 10 (1975), 729. doi: 10.2977/prims/1195191889. Google Scholar [25] P. Laurençot and C. Walker, A stationary free boundary problem modeling electrostatic MEMS,, Arch. Rat. Mech. Anal., 207 (2013), 139. doi: 10.1007/s00205-012-0559-7. 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] C. Liang, J. Li and K. Zhang, On a hyperbolic equation arising in electrostatic MEMS,, J. Differential Equations, 256 (2014), 503. doi: 10.1016/j.jde.2013.09.010. Google Scholar [28] C. Liang and K. Zhang, Asymptotic stability and quenching behavior of a hyperbolic nonlocal MEMS equation,, Commun. Math. Sci., 13 (2015), 355. doi: 10.4310/CMS.2015.v13.n2.a5. Google Scholar [29] J. A. Pelesko and A. A. Bernstein, Modeling MEMS and NEMS,, Chapman and Hall, (2003). Google Scholar [30] R. A. Smith, On a hyperbolic quenching problem in several dimensions,, SIAM J. Math. Anal., 20 (1989), 1081. doi: 10.1137/0520072. Google Scholar [31] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 37 (2010), 259. doi: 10.1007/s00526-009-0262-1. Google Scholar
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