Quenching behaviour of a nonlocal parabolic MEMS equation

Abstract

We obtain upper bounds for the quenching time of the solutions of the nonlocal parabolic MEMS equation $u_t=\Delta u+\lam/(1-u)^2(1+\chi\int_{\Omega}1/(1-u) dx)^2$ in $\Omega\times (0,\infty)$, $u=0$ on $\1\Omega\times (0,\infty)$, $u(x,0)=u_0$ in $\Omega$, when $\lambda$ is large. We prove the compactness of the quenching set under a mild condition on the initial data. When $\Omega=B_R$ and $u_0$ is radially symmetric and monotone decreasing in $0\le r\le R$, we prove that the point $x=0$ is the only possible quenching set. When $u_0$ also satisfies some strict concavity assumption, we prove that for any $\beta\in (2,3)$ the solution satisfies $1-u(x,t)\ge C|x|^{\frac{2}{\beta}}$ for some constant $C>0$ and we also obtain the quenching time estimate in this case.