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

Abstract

We consider a relaxation of the viscous Cahn-Hilliard equation induced by the second-order inertial term utt. 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 utt, the term f(u) is diffcult 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.

Keywords

• Cahn-Hilliard equation| inertia| weak formulation| maximal monotone operator| duality