STABILITY OF EXPONENTIAL ATTRACTORS FOR SINGULARLY PERTURBED PHASE-FIELD SYSTEMS OF OONO TYPE

We consider a phase-field system that takes into account the long-ranged interactions in phase separation based on a theory introduced by Y. Oono et al. (Formula presented), where (Formula presented) is the heat capacity, ϕ is the order parameter, u is the absolute temperature, and the Laplace operator N = −∆: D(N) →^˙L²(Ω) is subject to either Neumann boundary conditions (in which case (Formula presented) is a bounded domain with smooth boundary) or periodic boundary conditions (in which case Ω = Π^di=1^{(0, L}i), L_i > 0), d = 1, 2 or 3. We consider a class of nonlinear functions (Formula presented) that contains the polynomial (Formula presented). We prove a well-posedness result and the existence of the global attractor which is upper semicontinuous at ε = 0. Then we construct a family of exponential attractors that is Hölder continuous at ε = 0. Our present contribution completes and generalizes some recent results proven by Bonfoh and Suleman in [Comm. Pure Appl. Anal. 2021; 20: 3655-3682] where a conserved model (corresponding to σ = γ = 0 in (S_ε)) that takes into account the viscosity effects in the material was considered.