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

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 $d\le 2$. 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.