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

We study a generalized Caginalp phase-field system based on the theory of type III heat conduction proposed by Green and Naghdi and supplemented with Neumann boundary conditions. In contrast to the Dirichlet case, the system exhibits a lack of dissipation on the thermal displacement variable α. However, α minus its spatial average is dissipative and we are able to prove the existence of the global attractor with optimal regularity for the associated semigroup