The paper investigates a hyperbolic relaxation of the viscous Cahn–Hilliard system on a bounded, smooth domain (Omegasubsetmathbb{R}^{N}) with (N=1,2,3). The model couples a wave‑type equation for the chemical potential (mu) with a parabolic equation for the order parameter (phi). The system reads
[
alpha,partial_{tt}mu+partial_{t}phi-Deltamu=0,qquad
tau,partial_{t}phi-Deltaphi+f'(phi)=mu+g,
]
subject to homogeneous Neumann boundary conditions and prescribed initial data. The new feature is the second time derivative (alpha,partial_{tt}mu); the parameter (alpha>0) is a relaxation coefficient that will later be sent to zero. The viscosity coefficient (tau>0) appears in the second equation, while (g) is a known forcing term that can be interpreted as a control input. The potential (f) is split into a possibly non‑differentiable convex part (f_{1}) and a smooth concave perturbation (f_{2}). Three physically relevant choices are considered: the regular quartic potential, the logarithmic double‑well potential, and the double‑obstacle potential. In the logarithmic case the derivative (f'(phi)) becomes singular at (phi=pm1), forcing the solution to stay in the interval ((-1,1)); for the double‑obstacle case the subdifferential of the indicator function of ([-1,1]) replaces the derivative of (f_{1}).

The authors establish a comprehensive well‑posedness theory. Existence of a weak solution is proved by a two‑step approximation: first a Yosida regularisation of the subdifferential of (f_{1}) is introduced, then a Faedo–Galerkin scheme is applied. The analysis yields a priori estimates that are uniform with respect to the regularisation parameters, allowing passage to the limit and the construction of a solution satisfying the original system. Uniqueness follows from a standard continuous‑dependence argument, which also provides stability with respect to perturbations of the data. Regularity results show that (phi) belongs to (H^{1}(0,T;H^{1}(Omega))) and that (mu) has additional spatial regularity depending on the smoothness of the potential.

A key contribution is the asymptotic analysis as (alphato0). The authors prove that solutions of the relaxed system converge to solutions of the classical viscous Cahn–Hilliard equations. Moreover, they derive an explicit convergence rate: the difference between the relaxed and the limit solutions is bounded by (C,alpha), where (C) depends only on the data and the final time. This result clarifies the role of the hyperbolic term and quantifies how fast the relaxation vanishes.

The work is part of a broader research programme on phase‑separation phenomena in alloys, building on earlier collaborations that applied Cahn–Hilliard models to tin/lead systems. The present study is supported by the MIUR‑PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)”. The authors, Pierluigi Colli of the University of Pavia and Jürgen Sprekels of the Weierstrass Institute (WIAS) in Berlin, exchanged visits during the project’s development. The preprint, numbered 3128 in the WIAS series, was submitted on 4 September 2024 and is dedicated to Prof. Dr. Wolfgang Dreyer. The collaboration combines expertise in nonlinear partial differential equations, functional analysis, and applied mathematics, and the results contribute to the theoretical understanding of hyperbolic relaxations in phase‑field models.