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November  2016, 15(6): 2447-2456. doi: 10.3934/cpaa.2016043

## On some touchdown behaviors of the generalized MEMS device equation

 1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received  March 2016 Revised  May 2016 Published  September 2016

We study the quenching behaviors for the generalized microelectromechanical system (MEMS) equation $u_{t}-\Delta u=\lambda\rho(x)f(u)$, $0 < u < A$ ($A=1$ or $+\infty$), in $\Omega\times(0,+\infty)$, $u(x,t)=0$ on $\partial\Omega\times(0,+\infty)$, $u(x,0)=u_{0}(x)\in[0,A)$ in $\Omega$, where $\lambda>0$, $\Omega\subset R^N$ is a bounded domain, $0\le \rho(x) \in C^{\alpha}(\overline{\Omega})$, $\rho\not\equiv0$, for some constant $0 < \alpha < 1$, $0 < f \in C^{2}((0,A))$ such that $f'(s)\ge0$, $f''(s)\ge0$ for any $s\in[0,A)$ and $u_{0}$ is smooth, $u_{0}=0$ on $\partial\Omega$. It is well known that quenching does occur and corresponds to a touchdown phenomenon. We establish an interesting quenching rate, and based on which we then prove that touchdown cannot occur at zero points of $\rho(x)$ or at the boundary of $\Omega$, without the assumption of compactness of the touchdown set.
Citation: Qi Wang. On some touchdown behaviors of the generalized MEMS device equation. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2447-2456. doi: 10.3934/cpaa.2016043
##### References:
 [1] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.  Google Scholar [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 Applied Math, 60 (2005), 1731-1768. doi: 10.1002/cpa.20189.  Google Scholar [3] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lect. Notes in Math, 20, Courant Institute of Mathematical Sciences, New York University, New York, (2010). doi: 10.1090/cln/020.  Google Scholar [4] A. Friedman and B. Mcleod, Blow up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [5] 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.  Google Scholar [6] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case, NoDEA Nonlinear Differ. Equ. Appl., 15 (2008), 115-145. doi: 10.1007/s00030-007-6004-1.  Google Scholar [7] 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.  Google Scholar [8] J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal., 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9.  Google Scholar [9] J.-S. Guo and P. Souplet, No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal., 47 (2015), 614-625. doi: 10.1137/140970070.  Google Scholar [10] Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, J. Differential Equations, 224 (2008), 2277-2309. doi: 10.1016/j.jde.2008.02.005.  Google Scholar [11] Y. Guo, Z. G. 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.  Google Scholar [12] K. M. Hui, Global and touchdown behaviour of the generalized MEMS, Adv. Math. Sci. Appl., 19 (2009), 347-370.  Google Scholar [13] N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Differ. Equ. Appl., 15 (2008), 363-385. doi: 10.1007/s00030-008-7081-5.  Google Scholar [14] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.  Google Scholar [15] P. Quittner and PH. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Birkhäuser, Basel, 2007.  Google Scholar [16] J. A. Pelesko and D. H. Berstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, (2002). Google Scholar [17] Q. Wang, Dynamical solutions of singular parabolic equations modeling electrostatic MEMS, Nonlinear Differ. Equ. Appl., 22 (2015), 629-650. doi: 10.1007/s00030-014-0298-6.  Google Scholar [18] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 37 (2010), 259-274. doi: 10.1007/s00526-009-0262-1.  Google Scholar

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##### References:
 [1] H. Brezis, T. Cazenave, Y. Martel and A. Ramiandrisoa, Blow up for $u_t-\Delta u=g(u)$ revisited, Adv. Differential Equations, 1 (1996), 73-90.  Google Scholar [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 Applied Math, 60 (2005), 1731-1768. doi: 10.1002/cpa.20189.  Google Scholar [3] P. Esposito, N. Ghoussoub and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lect. Notes in Math, 20, Courant Institute of Mathematical Sciences, New York University, New York, (2010). doi: 10.1090/cln/020.  Google Scholar [4] A. Friedman and B. Mcleod, Blow up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar [5] 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.  Google Scholar [6] N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: dynamic case, NoDEA Nonlinear Differ. Equ. Appl., 15 (2008), 115-145. doi: 10.1007/s00030-007-6004-1.  Google Scholar [7] 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.  Google Scholar [8] J.-S. Guo, On the quenching behavior of the solution of a semilinear parabolic equation, J. Math. Anal., 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9.  Google Scholar [9] J.-S. Guo and P. Souplet, No touchdown at zero points of the permittivity profile for the MEMS problem, SIAM J. Math. Anal., 47 (2015), 614-625. doi: 10.1137/140970070.  Google Scholar [10] Y. Guo, On the partial differential equations of electrostatic MEMS devices III: refined touchdown behavior, J. Differential Equations, 224 (2008), 2277-2309. doi: 10.1016/j.jde.2008.02.005.  Google Scholar [11] Y. Guo, Z. G. 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.  Google Scholar [12] K. M. Hui, Global and touchdown behaviour of the generalized MEMS, Adv. Math. Sci. Appl., 19 (2009), 347-370.  Google Scholar [13] N. I. Kavallaris, T. Miyasita and T. Suzuki, Touchdown and related problems in electrostatic MEMS device equation, Nonlinear Differ. Equ. Appl., 15 (2008), 363-385. doi: 10.1007/s00030-008-7081-5.  Google Scholar [14] J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.  Google Scholar [15] P. Quittner and PH. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Adv. Texts, Birkhäuser, Basel, 2007.  Google Scholar [16] J. A. Pelesko and D. H. Berstein, Modeling MEMS and NEMS, Chapman & Hall/CRC, (2002). Google Scholar [17] Q. Wang, Dynamical solutions of singular parabolic equations modeling electrostatic MEMS, Nonlinear Differ. Equ. Appl., 22 (2015), 629-650. doi: 10.1007/s00030-014-0298-6.  Google Scholar [18] D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differential Equations, 37 (2010), 259-274. doi: 10.1007/s00526-009-0262-1.  Google Scholar
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