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Asymptotic Behaviors of Global Solutions to Fourth-order Parabolic and Hyperbolic Equations with Dirichlet Boundary Conditions

Wenlong Wu, Yanyan Zhang

Abstract

This paper investigates the asymptotic behaviors of global solutions to fourth-order parabolic and hyperbolic equations with Dirichlet boundary conditions. The equations model Micro-Electro-Mechanical Systems (MEMS) and are depending on a positive voltage parameter $λ$. We establish the convergence of global solutions to an equilibrium, along with the convergence rate estimates. Supporting numerical simulations are presented.

Asymptotic Behaviors of Global Solutions to Fourth-order Parabolic and Hyperbolic Equations with Dirichlet Boundary Conditions

Abstract

This paper investigates the asymptotic behaviors of global solutions to fourth-order parabolic and hyperbolic equations with Dirichlet boundary conditions. The equations model Micro-Electro-Mechanical Systems (MEMS) and are depending on a positive voltage parameter . We establish the convergence of global solutions to an equilibrium, along with the convergence rate estimates. Supporting numerical simulations are presented.
Paper Structure (16 sections, 22 theorems, 366 equations, 7 figures)

This paper contains 16 sections, 22 theorems, 366 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic Problem
  4. Gradient system
  5. Lojasiewicz-Simon inequality
  6. Convergence rate
  7. Hyperbolic Problem
  8. Energy Dissipation Identity
  9. Gradient System
  10. Lojasiewicz-Simon inequality
  11. Proof of Theorem \ref{['theorem1.3']}
  12. Comparison with parabolic MEMS equation
  13. Numerical Simulations
  14. Numerical Solutions for the Parabolic MEMS Equation
  15. Numerical Solutions for the Hyperbolic MEMS Equation
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^{d}$($d=1,2$) be an arbitrary bounded smooth domain. Let $B>0$, $T\ge 0$, $\kappa\in(0,1)$, $0<\lambda <\frac{\kappa^{2}(8-3\kappa)}{128C_{0}^{2}|\Omega|}$. For any given $u_{0}\in X(\kappa)$, if the solution $u$ to neweq1.1 globally exists and satisfies $u\in X(\kappa) Furthermore, there is $C_{1}>0$, $\gamma_{1}>0$, and $T_{1}>0$ such that when $t\ge T_{1}$,

Figures (7)

  • Figure 1: A simplified MEMS device
  • Figure 2: Plot of the solution $u(t,x)$ for the parabolic MEMS equation \ref{['neweq1.1']} at fixed $t$
  • Figure 3: Plot of the solution $u(t,x)$ for the parabolic MEMS equation \ref{['neweq1.1']} at fixed $x=0$
  • Figure 4: Plot of the solution $u(t,x)$ for the parabolic MEMS equation \ref{['neweq1.1']} at fixed $(t,x)$
  • Figure 5: Plot of the solution $u(t,x)$ for the hyperbolic MEMS equation \ref{['neweq1.2']} at fixed $t$
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: ref3
  • Definition 2.2: ref3
  • Theorem 2.1: ref3
  • Definition 2.3
  • Theorem 2.2: ref10
  • Definition 2.4: ref5
  • Theorem 2.3: ref3
  • Theorem 2.4: ref5
  • ...and 31 more