Quenching Phenomenon for a Parabolic MEMS Equation


Qi WANG.Quenching Phenomenon for a Parabolic MEMS Equation[J].Chinese Annals of Mathematics B,2018,39(1):129~144
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Abstract: This paper deals with the electrostatic MEMS-device parabolic equation $$u_{t}-\Delta u=\d\frac{\lambda f(x)}{(1-u)^{p}}$$ in a bounded domain $\O$ of $\r^{N}$, with Dirichlet boundary condition, an initial condition $u_{0}(x)\in[0,1)$ and a nonnegative profile $f$, where $\la>0$, $p>1$. The study is motivated by a simplified micro-electromechanical system (MEMS for short) device model. In this paper, the author first gives an asymptotic behavior of the quenching time $T^{*}$ for the solution $u$ to the parabolic problem with zero initial data. Secondly, the author investigates when the solution $u$ will quench, with general $\la$, $u_{0}(x)$. Finally, a global existence in the MEMS modeling is shown.


MEMS equation, Quenching time, Global existence


35A01, 35B44, 35K58
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