MATHEMATICAL ANALYSIS OF NONLINEAR DIFFERENTIAL EQUATION ARISING IN MEMS

Title & Authors
MATHEMATICAL ANALYSIS OF NONLINEAR DIFFERENTIAL EQUATION ARISING IN MEMS
Zhang, Ruifeng; Li, Na;

Abstract
In this paper, we study nonlinear equation arising in MEMS modeling electrostatic actuation. We will prove the local and global existence of solutions of the generalized parabolic MEMS equation. We present that there exists a constant $\small{{\lambda}^*}$ such that the associated stationary problem has a solution for any $\small{{\lambda}}$ < $\small{{\lambda}^*}$ and no solution for any $\small{{\lambda}}$ > $\small{{\lambda}^*}$. We show that when $\small{{\lambda}}$ < $\small{{\lambda}^*}$ the global solution converges to its unique maximal steady-state as $\small{t{\rightarrow}{\infty}}$. We also obtain the condition for the existence of a touchdown time $\small{T{\leq}{\infty}}$ for the dynamical solution. Furthermore, there exists $\small{p_0}$ > 1, as a function of $\small{p}$, the pull-in voltage $\small{{\lambda}^*(p)}$ is strictly decreasing with respect to 1 < $\small{p}$ < $\small{p_0}$, and increasing with respect to $\small{p}$ > $\small{p_0}$.
Keywords
MEMS equation;upper-and-lower solution method;global convergence;touchdown time;
Language
English
Cited by
References
1.
P. Esposito, N. Ghoussoub, and Y. Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics 20, New York, 2010.

2.
G. Flores, G. A. Mercado, J. A. Pelesko, and N. Smyth, Analysis of the dynamics and touchdown in a model of electrostatic MEMS, SIAM J. Appl. Math. 67 (2006/2007), no. 2, 434-446.

3.
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewood Cliffs, N.J., U.S.A., 1964.

4.
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices: Stationary case, SIAM J. Math. Anal. 38 (2006/07), no. 5, 1423-1449.

5.
N. Ghoussoub and Y. Guo, On the partial differential equations of electrostatic MEMS devices II: Dynamic case, Nonlinear Differential Equations Appl. 15 (2008), no. 1-2, 115-145.

6.
Y. Guo, Z. Pan, and M. Ward, Touchdown and pull-in voltage behavior of a MEMS device with varying dielectric properties, SIAM J. Appl. Math. 66 (2005), no. 1, 309- 338.

7.
K. M. Hui, Global and touchdown behaviour of the generalized MEMS device equation, Adv. Math. Sci. Appl. 19 (2009), no. 2, 347-370.

8.
K. M. Hui, Quenching behaviour of a nonlocal parabolic MEMS equation, http:// arxiv.org/0908.1227v2.

9.
O. A. Ladyzenskaja, V. A. Solonnikov, and N. N. Uralceva, Linear and Quasilinear Equations Of Parabolic Type, Transl. Math. Monographs, Amer. Math. Soc., 23, Providence, R. I. USA, 1968.

10.
F. H. Lin and Y. S. Yang, Nonlinear non-local elliptic equation modelling electrostatic actuation, Proc. R. Soc. A 463 (2007), no. 2081, 1323-1337.

11.
J. A. Pelesko, Mathematical modeling of electrostatic MEMS with tailored dielectric properties, SIAM J. Appl. Math. 62 (2001/2002), no. 3, 888-908.

12.
Z. P. Wang and L. Z. Ruan, On a class of semilinear elliptic problems with singular nonlinearities, Appl. Math. Comput. 193 (2007), no. 1, 89-105.

13.
D. Ye and F. Zhou, On a general family of nonautonomous elliptic and parabolic equations, Calc. Var. Partial Differential Equations 37 (2010), no. 1-2, 259-274.