Attractors and Numerical Simulations for a Two-Temperature Phase Transition System

In this paper, we study a generalisation of the Caginalp phase field system obtained by considering the Fourier law as the heat conduction law and involving two temperatures. In the first section, we are interested in the existence and uniqueness of solutions. In section 2, we address the question of the asymptotic behaviour of the solutions, in particular with the demonstration of the existence of a global attractor. For this, we needed to show that the system is dissipative and to know the regularity of the solutions. Finally, in the last section, we turn to the numerical solution of our problem.