# American Institute of Mathematical Sciences

August  2013, 18(6): 1581-1610. doi: 10.3934/dcdsb.2013.18.1581

## Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions

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

Received  October 2011 Revised  December 2011 Published  March 2013

We consider a modified Cahn-Hiliard equation where the velocity of the order parameter $u$ depends on the past history of $\Delta \mu$, $\mu$ being the chemical potential with an additional viscous term $\alpha u_{t},$ $\alpha >0.$ In addition, the usual no-flux boundary condition for $u$ is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The aim of this work is to analyze the passage to the singular limit when the memory kernel collapses into a Dirac mass. In particular, we discuss the convergence of solutions on finite time-intervals and we also establish stability results for global and exponential attractors.
Citation: Ciprian G. Gal, Maurizio Grasselli. Singular limit of viscous Cahn-Hilliard equations with memory and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1581-1610. doi: 10.3934/dcdsb.2013.18.1581
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##### References:
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