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

Yufei Wei, Yanyan Zhang

Abstract

We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-Δu=\fracλ{(1-u)^2\bigl(1+\int_Ω\frac{1}{1-u}\,dx\bigr)^2},\quad x\inΩ,\ t>0, \] where \(Ω\subset\mathbb{R}^N\) \((1\le N\le3)\) is a bounded smooth domain and \(λ>0\). Using operator semigroups and the contraction mapping principle, we prove local existence and give a quenching criterion. Under suitable smallness conditions on \(λ\) 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 \(λ\)-dichotomy.

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

Abstract

We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: where \((1\le N\le3)\) is a bounded smooth domain and . Using operator semigroups and the contraction mapping principle, we prove local existence and give a quenching criterion. Under suitable smallness conditions on 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 , 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 -dichotomy.
Paper Structure (25 sections, 25 theorems, 238 equations, 18 figures)

This paper contains 25 sections, 25 theorems, 238 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Research background
  3. Research status
  4. Main conclusion
  5. Preliminaries
  6. Contraction mapping principle
  7. operator semigroup
  8. Analytic function and analytic mapping
  9. Semi-Fredholm Operators
  10. Lojasiewicz Inequality
  11. Steady State equation
  12. Local existence
  13. Global existence
  14. Asymptotic behavior of solutions
  15. Gradient system
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^{N}$ and $u_0 \in H^{2}\cap H_{0}^{1}(\Omega)$, $\|u_{0}\|_{L^{\infty}(\Omega)} \le 1-2 \delta$, where $\delta \in\left(0, \frac{1}{2}\right)$, and $u_0$ is continuous on $\bar{\Omega}$. Then for any $\lambda>0$, there exists $\widetilde{T}>0$ depending on $\Omega$, $\de Moreover, there exists $T^{*}>0$ such that $[0,T^{*})$ is the maximal existence interval, and eithe

Figures (18)

  • Figure 1: MEMS device(ref7)
  • Figure 2: capacitive control circuit (ref15)
  • Figure 3: Solution of equation (\ref{['3.1']}) for $\lambda=8.53$ in $t\in[0,10]$
  • Figure 4: Solution of equation (\ref{['3.1']}) for $\lambda=8.54$ in $t\in[0,8.5]$
  • Figure 5: Solution values at $x=0.5$ for $\lambda=8$ to $9$ in equation (\ref{['3.1']})
  • ...and 13 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.1
  • Theorem 1.5
  • Theorem 2.1: rud
  • Definition 2.1: See Definition 2.1 in ref18
  • Definition 2.2: See p. 238 in ref8
  • Definition 2.3: are
  • ...and 41 more