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On some touchdown behaviors of the generalized MEMS device equation

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


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