On the viscous Cahn-Hilliard equation with singular potential and inertial term

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term u_tt. The equation also contains a semilinear term f(u) of "singular" type. Namely, the function f is defined only on a bounded interval of R corresponding to the physically admissible values of the unknown u, and diverges as u approaches the extrema of that interval. In view of its interaction with the inertial term u_tt, the term f(u) is difficult to be treated mathematically. Based on an approach originally devised for the strongly damped wave equation, we propose a suitable concept of weak solution based on duality methods and prove an existence result.