Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations

TL;DR

This paper investigates the long-term behavior of solutions to certain fourth-order nonlinear PDEs, establishing the existence of global and exponential attractors, and analyzing their convergence properties in specific limits.

Contribution

It proves the existence of global and exponential attractors with finite fractal dimensions for classes of fourth-order nonlinear PDEs, including the LLBar and CH-AC equations, and studies their convergence in limiting cases.

Findings

Existence of global and exponential attractors with finite fractal dimensions.

Convergence of exponential attractors to those of limiting equations in specific parameter limits.

Rate estimates for convergence to stationary states in certain cases.

Abstract

We study the long-time behaviour of solutions to some classes of fourth-order nonlinear PDEs with non-monotone nonlinearities, which include the Landau--Lifshitz--Baryakhtar (LLBar) equation (with all relevant fields and spin torques) and the convective Cahn--Hilliard/Allen--Cahn (CH-AC) equation with a proliferation term, in dimensions $d=1,2,3$. Firstly, we show the global well-posedness, as well as the existence of global and exponential attractors with finite fractal dimensions for these problems. In the case of the exchange-dominated LLBar equation and the CH-AC equation without convection, an estimate for the rate of convergence of the solution to the corresponding stationary state is given. Finally, we show the existence of a robust family of exponential attractors when $d\leq 2$. As a corollary, exponential attractor of the LLBar equation is shown to converge to that of the Landau--Lifshitz--Bloch equation in the limit of vanishing exchange damping, while exponential attractor of the convective CH-AC equation is shown to converge to that of the convective Allen--Cahn equation in the limit of vanishing diffusion coefficient.