Monotonicity conjecture for solutions of 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 $u_0\in H^2\cap H_0^1(\Omega)$. Let $\lambda^{*}$ be the critical parameter from the critical-parameter dichotomy conjecture, and for $\lambda<\lambda^{*}$ let $u_\lambda$ denote the corresponding solution of equation (1.1). Monotonicity conjecture. For every finite $t$, the map

$$\lambda\longmapsto u_\lambda(\cdot,t)$$

is monotonically increasing in $H^2\cap H_0^1(\Omega)$. This conjecture formalizes the monotone dependence on $\lambda$ suggested by the numerical experiments for the global-solution regime.