Mathematical analysis of nonlinear differential equation arising in MEMS
Bull. Korean Math. Soc. 2012 Vol. 49, No. 4, 705-714
https://doi.org/10.4134/BKMS.2012.49.4.705
Published online July 1, 2012
Ruifeng Zhang and Na Li
Henan University, Wan fang College of Science Technology HPU
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 $\lambda^{*}$ such that the associated stationary problem has a solution for any $\lambda < \lambda^{*}$ and no solution for any $\lambda > \lambda^{*}$. We show that when $\lambda < \lambda^{*}$ the global solution converges to its unique maximal steady-state as $t\rightarrow\infty$. We also obtain the condition for the existence of a touchdown time $T\leq\infty$ for the dynamical solution. Furthermore, there exists $p_0>1$, as a function of $p$, the pull-in voltage $\lambda^{*}(p)$ is strictly decreasing with respect to $1p_0$.
Keywords : MEMS equation, upper-and-lower solution method, global convergence, touchdown time
MSC numbers : 35K55, 35K65, 35B40
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