The Cahn-Hilliard/Allen-Cahn equation with inertial and proliferation terms

This paper is concerned with the initial-boundary value problem for the hyperbolic relaxation of the Cahn-Hilliard/Allen-Cahn equation with a proliferation term, in an arbitrary two dimensional smooth domain. With appropriate assumptions on the nonlinearities, we first prove the existence and uniqueness of an energy solution of the considered problem. Then, establishing the validity of the energy equality we show the regularity (in time) of the energy solution and its continuous dependence on the initial data. Under additional conditions on the nonlinearities, we also prove that the associated semigroup possesses a global attracto