Some dynamical properties of constrained Modified Swift-Hohenberg Equation

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.