A generalization of the Caginalp phase-field system with Neumann boundary conditions (Q393207)

This paper studies the phase-field system NEWLINE\[NEWLINEu_t-\Delta u +\phi(u)=\alpha_tNEWLINE\]NEWLINE NEWLINE\[NEWLINE\alpha_t-\Delta \alpha_t-\Delta \alpha =-u_t,NEWLINE\]NEWLINE in which \(u\) is an order parameter and \(\alpha_t\) is temperature. This system is a model for solid-liquid phase transition, in which heat conduction follows a law of type III. The system is considered with Neumann boundary conditions on a smooth bounded domain in \(\mathbb R^3\). Under appropriate conditions on the nonlinearity \(\phi\), existence of a compact global attractor for the evolution in the appropriate functional space is proved.