Global existence for a strongly coupled Cahn-Hilliard system with viscosity

An existence result is proved for a nonlinear diffusion problem of phase-field type, consisting of a parabolic system of two partial differential equations, complemented by Neumann homogeneous boundary conditions and initial conditions. This system is meant to model two-species phase segregation on an atomic lattice under the presence of diffusion. A similar system has been recently introduced and analyzed in the paper arXiv:1103.4585 . Both systems conform to the general theory developed in [P. Podio-Guidugli, Models of phase segregation and diffusion of atomic species on a lattice, Ric. Mat. 55 (2006) 105-118]: two parabolic PDEs, interpreted as balances of microforces and microenergy, are to be solved for the order parameter and the chemical potential. In the system studied in this note, a phase-field equation fairly more general than in arXiv:1103.4585 is coupled with a highly nonlinear diffusion equation for the chemical potential, in which the conductivity coefficient is allowed to depend nonlinearly on both variables.