Some dynamical properties of constrained Modified Swift-Hohenberg Equation

TL;DR

This paper investigates the long-term dynamics of a constrained modified Swift-Hohenberg equation, demonstrating convergence to equilibrium, analyzing convergence rates, and establishing the existence of a global attractor.

Contribution

It introduces a detailed analysis of the constrained modified Swift-Hohenberg equation's long-term behavior, including convergence and attractor existence, using Lojasiewicz-Simon inequality.

Findings

Solutions tend to equilibrium states over time.

The rate of convergence to equilibrium is characterized.

A global attractor for the system is proven to exist.

Abstract

In this paper, we have studied the long-term behavior for the projected deterministic constrained modified Swift-Hohenberg equation with constraints and Dirichlet boundary conditions. Specifically, using Lojasiewicz-Simon inequality, we have shown that the global solution approaches an equilibrium state. Also, we have analyzed the rate at which the solution approaches equilibrium. Finally, we have proven the existence of a global attractor.