Phase-field system with two temperatures and a nonlinear coupling term based on the Maxwell-Cattaneo law

We study a generalisation of the Caginalp phase field system derived from the Maxwell-Cattaneo law on the one hand, and from the two-temperature theory of heat conduction on the other. We first show that the problem is well posed and obtain dissipativity properties of the associated solution operators. We then prove estimates \({\textbf{L}^\infty }\) ${\mathbf{L}}^{\infty }$ -estimates relying on regularity estimates based on Moser’s iterations. We finally analyze a spatial behaviour of the solution in a semi-infinite cylinder, when such solutions exist.