On the quenching behavior of the MEMS with fringing field (Q2788679)

The authors study a quenching phenomenon for the singular parabolic equation NEWLINE\[NEWLINEu_{t}-\Delta u=\lambda \frac{1+\delta \left| \nabla u\right| ^{2}}{ \left( 1-u\right) ^{2}},NEWLINE\]NEWLINE which models the behavior of micromechanical systems. The problem is posed in \(\Omega \times (0,T)\) with \(\Omega \subset \mathbb{R}^{n}\) and homogeneous Dirichlet boundary conditions and initial data are imposed. The authors make precise earlier results obtained by \textit{Q. Wang} in [J. Math. Anal. Appl. 405, No. 1, 135--147 (2013; Zbl 1515.35135)] who proved the existence of a pull-in voltage \(\lambda _{\delta }^{\ast }\) such that, if \(\lambda \leq \lambda _{\delta }^{\ast }\), there exists a unique global solution to the above problem which converges onotonically and pointwise as \(t\rightarrow \infty \) to the unique solution of the stationary associated equation NEWLINE\[NEWLINE-\Delta u=\lambda \frac{1+\delta \left| \nabla u\right| ^{2}}{\left( 1-u\right) ^{2}}NEWLINE\]NEWLINE with \(u=0\) on \(\partial \Omega \) , and, if \(\lambda >\lambda _{\delta }^{\ast }\), the unique solution of the non-steady problem must quench in finite time. In the present paper, the authors first prove that the quenching time \(T(t,\delta )\) satisfies \(\lim \sup_{\lambda \rightarrow \infty }\lambda T=\frac{1}{3}\). Moreover, if \( \Omega \) is the ball \(B_{R}\), for \(\lambda >\lambda _{\delta }^{\ast }\), the solution to the non-steady problem only quenches at \(r=0\). The proof of the existence result is first based on the transformation NEWLINENEWLINE\[NEWLINE\zeta _{\lambda ,\delta }(u(x,t))=\int_{0}^{u(x,t)}e^{\frac{\lambda s}{1-s}}ds,NEWLINE\]NEWLINE which leads to the nonlinear equation \(v_{t}-\Delta v=\lambda \rho _{\lambda ,\delta }(v) \) with a nonlinear term \(\rho _{\lambda ,\delta }\) whose properties are studied. Then, the authors use a maximum principle and an existence result for the stationary equation. For the quenching result in finite time, the authors use a comparison principle with a sub- and super-solution argument, arguing by contradiction and the preceding transformation. They prove estimates on the pull-in voltage and on the quenching time through comparison arguments. They finally study the quenching set in the cases of a convex domain and of a ball. The paper ends with a presentation of numerical simulations.