Well-posedness and long-time behavior for a nonstandard viscous Cahn--Hilliard system

We study a diffusion model of phase field type, consisting {of} a system of two partial differential equations {encoding} the balance{s} of microforces and microenergy; the two unknowns are the order parameter and the chemical potential. By a careful development of uniform estimates and the deduction of {certain} useful boundedness properties, we prove existence and uniqueness of a global-in-time smooth solution to the {associated} initial/boundary-value problem; moreover, we give a description of the relative $\omega$-limit set