Dynamics of the viscous Cahn-Hilliard equation |
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Authors: | AN Carvalho Tomasz Dlotko |
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Institution: | a Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil b Institute of Mathematics, Silesian University, 40-007 Katowice, Poland |
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Abstract: | We study generalized viscous Cahn-Hilliard problems with nonlinearities satisfying critical growth conditions in , where Ω is a bounded smooth domain in Rn, n?3. In the critical growth case, we prove that the problems are locally well posed and obtain a bootstrapping procedure showing that the solutions are classical. For p=2 and almost critical dissipative nonlinearities we prove global well posedness, existence of global attractors in and, uniformly with respect to the viscosity parameter, L∞(Ω) bounds for the attractors. Finally, we obtain a result on continuity of regular attractors which shows that, if n=3,4, the attractor of the Cahn-Hilliard problem coincides (in a sense to be specified) with the attractor for the corresponding semilinear heat equation. |
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Keywords: | Viscous Cahn-Hilliard equation Global attractor Attractors Lower semicontinuity |
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