Global attractor and robust exponential attractors for some classes of fourth-order nonlinear evolution equations
Beniamin Goldys, Agus L. Soenjaya and Thanh Tran
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 . 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 . 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.