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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
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show all references

References:
[1]

Adv. Differential Equations, 1 (1996), 73-90.  Google Scholar

[2]

Comm. Pure Applied Math, 60 (2005), 1731-1768. doi: 10.1002/cpa.20189.  Google Scholar

[3]

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]

Indiana Univ. Math. J., 34 (1985), 425-447. doi: 10.1512/iumj.1985.34.34025.  Google Scholar

[5]

SIAM J. Math. Anal., 38 (2007), 1423-1449. doi: 10.1137/050647803.  Google Scholar

[6]

NoDEA Nonlinear Differ. Equ. Appl., 15 (2008), 115-145. doi: 10.1007/s00030-007-6004-1.  Google Scholar

[7]

Methods Appl. Anal., 15 (2008), 361-376. doi: 10.4310/MAA.2008.v15.n3.a8.  Google Scholar

[8]

J. Math. Anal., 151 (1990), 58-79. doi: 10.1016/0022-247X(90)90243-9.  Google Scholar

[9]

SIAM J. Math. Anal., 47 (2015), 614-625. doi: 10.1137/140970070.  Google Scholar

[10]

J. Differential Equations, 224 (2008), 2277-2309. doi: 10.1016/j.jde.2008.02.005.  Google Scholar

[11]

SIAM J. Appl. Math., 66 (2005), 309-338. doi: 10.1137/040613391.  Google Scholar

[12]

Adv. Math. Sci. Appl., 19 (2009), 347-370.  Google Scholar

[13]

Nonlinear Differ. Equ. Appl., 15 (2008), 363-385. doi: 10.1007/s00030-008-7081-5.  Google Scholar

[14]

SIAM J. Appl. Math., 62 (2002), 888-908. doi: 10.1137/S0036139900381079.  Google Scholar

[15]

Birkhäuser Adv. Texts, Birkhäuser, Basel, 2007.  Google Scholar

[16]

Chapman & Hall/CRC, (2002). Google Scholar

[17]

Nonlinear Differ. Equ. Appl., 22 (2015), 629-650. doi: 10.1007/s00030-014-0298-6.  Google Scholar

[18]

Calc. Var. Partial Differential Equations, 37 (2010), 259-274. doi: 10.1007/s00526-009-0262-1.  Google Scholar

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