Well-posedness and asymptotic behavior of solutions to a second order nonlocal parabolic MEMS equation

TL;DR

This paper studies a second-order nonlocal parabolic MEMS equation, proving well-posedness, quenching criteria, and long-term behavior, including convergence to steady states with rates depending on the Lojasiewicz exponent.

Contribution

It establishes local and global existence, quenching criteria, and convergence rates for solutions, using operator semigroups, energy analysis, and numerical experiments.

Findings

Global solutions exist under smallness conditions on parameters.

Solutions converge to steady states with exponential or algebraic rates.

Numerical experiments support theoretical results and conjectures.

Abstract

We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-\Delta u=\frac{\lambda}{(1-u)^2\bigl(1+\int_\Omega\frac{1}{1-u}\,dx\bigr)^2},\quad x\in\Omega,\ t>0, \] where \(\Omega\subset\mathbb{R}^N\) \((1\le N\le3)\) is a bounded smooth domain and \(\lambda>0\). Using operator semigroups and the contraction mapping principle, we prove local existence and give a quenching criterion. Under suitable smallness conditions on \(\lambda\) and the initial data, global existence and exponential convergence to the minimal steady state are obtained. Assuming the global solution stays uniformly away from the singularity \(u=1\), we show that the system forms a gradient system. By establishing analyticity of the energy and a Lojasiewicz--Simon inequality, we prove that the solution converges to a steady state with either exponential or algebraic rate depending on the Lojasiewicz exponent. Numerical experiments in 1D and 2D illustrate the results and support conjectures on the \(\lambda\)-dichotomy.