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March  2008, 7(2): 317-353. doi: 10.3934/cpaa.2008.7.317

Global attractors for a three-dimensional conserved phase-field system with memory

1. 

Dipartimento di Matematica "F.Brioschi", Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy

Received  January 2007 Revised  May 2007 Published  December 2007

We consider a conserved phase-field system on a tridimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature $\vartheta$. These effects are represented through a convolution integral whose relaxation kernel $k$ is a summable and decreasing function. Therefore the system consists of a linear integrodifferential equation for $\vartheta$ which is coupled with a viscous Cahn-Hilliard type equation governing the order parameter $\chi$. The latter equation contains a nonmonotone nonlinearity $\phi$ and the viscosity effects are taken into account by the term $-\alpha \Delta\chi_t$, for some $\alpha \geq 0$. Thus, we formulate a Cauchy-Neumann problem depending on $\alpha $. Assuming suitable conditions on $k$, we prove that this problem generates a dissipative strongly continuous semigroup $S^\alpha (t)$ on an appropriate phase space accounting for the past histories of $\vartheta$ as well as for the conservation of the spatial means of the enthalpy $\vartheta+\chi$ and of the order parameter. We first show, for any $\alpha \geq 0$, the existence of the global attractor $\mathcal A_\alpha $. Also, in the viscous case ($\alpha > 0$), we prove the finiteness of the fractal dimension and the smoothness of $\mathcal A_\alpha $.
Citation: Gianluca Mola. Global attractors for a three-dimensional conserved phase-field system with memory. Communications on Pure & Applied Analysis, 2008, 7 (2) : 317-353. doi: 10.3934/cpaa.2008.7.317
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