Critical-parameter dichotomy conjecture for the nonlocal parabolic MEMS equation

Let $\Omega\subset\mathbb{R}^N$, with $1\leq N\leq 3$, be a bounded domain with smooth boundary, and let $u_0\in H^2\cap H_0^1(\Omega)$. Consider equation (1.1), its solution $u_\lambda$, and the steady-state equation (1.4). Critical-parameter dichotomy conjecture. There exists $\lambda^{*}>0$ such that, for $\lambda<\lambda^{*}$, equation (1.1) has a unique global solution $u_\lambda$ converging to a steady state as $t\to\infty$, whereas, for $\lambda>\lambda^{*}$, the solution quenches in finite time and equation (1.4) has no solution. This conjecture proposes a threshold separating global convergence from finite-time quenching and nonexistence of steady states, based on the paper's numerical experiments.