A generalized Allen–Cahn model with mass source and its Cahn–Hilliard limit

The present paper is concerned with a fourth‐order Allen–Cahn model with logarithmic potential and mass source that describes the process of phase separation in two‐component systems accompanied by a flux of material. The existence of a global weak solution is obtained under appropriate hypotheses on the source term. Furthermore, we study its Cahn–Hilliard limit as a small parameter goes to zero. The main difficulty in the mathematical analysis of the model lies in the presence of the source term that leads to the nonconservation of mass, contrary to the original Cahn–Hilliard theory.