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

TL;DR

This paper studies the long-term behavior of solutions to fourth-order parabolic and hyperbolic equations modeling MEMS, proving convergence to equilibrium and providing rate estimates, supported by numerical simulations.

Contribution

It establishes the asymptotic convergence and rate estimates for solutions to MEMS-related equations, a novel analysis for these higher-order models.

Findings

Global solutions converge to equilibrium.

Convergence rates are explicitly estimated.

Numerical simulations support theoretical results.

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 $\lambda$. We establish the convergence of global solutions to an equilibrium, along with the convergence rate estimates. Supporting numerical simulations are presented.