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

Ciprian G. Gal; Maurizio Grasselli

doi:10.3934/dcdsb.2013.18.1581

Article Contents

2013, Volume 18, Issue 6: 1581-1610. Doi: 10.3934/dcdsb.2013.18.1581

This issue Previous Article Asymptotic behaviour for a class of delayed cooperative models with patch structure Next Article Parameter estimation by quasilinearization in differential equations with state-dependent delays

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

Ciprian G. Gal^1, and
Maurizio Grasselli^2,

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

Received: October 2011

Revised: December 2011

Published: August 2013

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Abstract

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.

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Mathematics Subject Classification: 35B41, 35Q99, 35R09, 37L30, 82C26.

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