  Existence and uniqueness are investigated 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 aims to model two-species phase segregation
on an atomic lattice; in the balance equations of microforces and microenergy,
the two unknowns are the order parameter and the chemical potential. A simpler
version of the same system has recently been discussed in arXiv:1103.4585v1. In
this paper, a fairly more general phase-field equation is coupled with a
genuinely nonlinear diffusion equation. The existence of a global-in-time
solution is proved with the help of suitable a priori estimates. In the case of
a constant atom mobility, a new and rather unusual uniqueness proof is given,
based on a suitable combination of variables.
