# American Institute of Mathematical Sciences

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January  2014, 34(1): 145-179. doi: 10.3934/dcds.2014.34.145

## Longtime behavior of nonlocal Cahn-Hilliard equations

 1 Department of Mathematics, Florida International University, Miami, FL, 33199 2 Dipartimento di Matematica, Politecnico di Milano, 20133 Milano

Received  July 2012 Revised  February 2013 Published  June 2013

Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.
Citation: Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145
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