On some touchdown behaviors of the generalized MEMS device equation

Qi  Wang

doi:10.3934/cpaa.2016043

Article Contents

2016, Volume 15, Issue 6: 2447-2456. Doi: 10.3934/cpaa.2016043

This issue Previous Article Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences Next Article Multiplicity and concentration of positive solutions for semilinear elliptic equations with steep potential

On some touchdown behaviors of the generalized MEMS device equation

Qi Wang^1,

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

Received: February 29, 2016

Revised: April 30, 2016

Published: October 2016

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Abstract

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.

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Mathematics Subject Classification: Primary: 35A01, 35B44; Secondary: 35K58.

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