Abstract.
We consider a singularly perturbed phase-field model of Caginalp type which is thermally isolated and whose order parameter \(\phi\) is subject to a dynamic boundary condition. More precisely, we indicate by ε a (small) coefficient multiplying ∂tu in the heat equation, u being the temperature, and we construct a family of exponential attractors which is robust as ε goes to 0. This is physically meaningful since the limiting problem is the viscous Cahn–Hilliard equation for the sole \(\phi\) with a dynamic boundary condition. The upper semicontinuity of the global attractor is also analyzed. The paper extends and revisits some results previously obtained by A. Miranville et al.
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Gal, C.G., Grasselli, M. & Miranville, A. Robust Exponential Attractors for Singularly Perturbed Phase-Field Equations with Dynamic Boundary Conditions. Nonlinear differ. equ. appl. 15, 535–556 (2008). https://doi.org/10.1007/s00030-008-7029-9
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DOI: https://doi.org/10.1007/s00030-008-7029-9