Profile of touch-down solution to a nonlocal MEMS model

Abstract

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0,T) \times \Omega:$ \begin{eqnarray*} \partial_t u = \Delta u +\displaystyle \frac{\lambda }{ (1-u)^2 \left( 1 +\displaystyle \gamma \int_{\Omega} \frac{1}{1-u} dx \right)^2}, \quad 0 \leq u <1, \end{eqnarray*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ and $\lambda, \gamma > 0$. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point $a \in \Omega$. In particular, we give a description of the quenching behavior according to the following final profile $$ 1 - u(x,T) \sim \theta^*\left[ \frac{|x-a|^2}{|\ln|x-a||} \right]^\frac{1}{3} \text{ as } x \to a, \theta^* > 0.$$ The construction relies on some connection between the quenching phemonenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The proof relies on two main steps: A reduction to a finite dimensional problem and a topological argument based on Index theory. The highlight of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.