On some touchdown behaviors of the generalized MEMS device equation
Qi Wang · Communications on Pure & Applied Analysis · 2016
We study the quenching behaviors for the generalized microelectromechanical system (MEMS) equation$u_{t}-\Delta u=\lambda\rho(x)f(u)$, $0 0$, $\Omega\subset R^N$ is a bounded domain, $0\le \rho(x) \in C^{\alpha}(\overline{\Omega})$,$\rho ot\equiv0$, 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 whichwe 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.