Critical-threshold conjecture for the hyperbolic MEMS equation

Assume $(B,T)$, $(u_{0},u_{1})$, $d$, and $\Omega$ are given, and let $u$ be the unique maximal solution to the hyperbolic problem. Critical-threshold conjecture. There exist critical values $0<\lambda_{h,1}^{*}<\lambda_{h,2}^{*}$ such that: for $0<\lambda<\lambda_{h,2}^{*}$, $u(t,x)$ exists globally; for $0<\lambda<\lambda_{h,1}^{*}$ and every fixed $(t,x)\in \mathbb{R}^{+}\times \Omega$, $u(t,x)$ is monotonically decreasing in $\lambda$; and for $\lambda>\lambda_{h,2}^{*}$, $u$ reaches the value $-1$ in finite time, with this time monotonically decreasing in $\lambda$. This conjecture proposes two thresholds separating global existence, parameter monotonicity, and finite-time touchdown; its resolution is not supplied here.