Robust exponential attractors for singularly perturbed conserved phase-field systems with no growth assumption on the nonlinear term

We consider the conserved phase-field system {equation presented} where τ > 0 is a relaxation time, δ > 0 is the viscosity parameter, ϵ ∈ (0, 1] is the heat capacity, φ is the order parameter, u is the absolute temperature, the Laplace operator N = -Δ: D(N) → L²(Ω) is subject to either Neumann boundary conditions (in which case Ω ⊂ R^dis a bounded domain with smooth boundary) or periodic boundary conditions (in which case Ω = Π^d_i=1(0, L_i), L_i> 0), d = 1, 2 or 3, and G(φ) = ∫^φ₀g(σ)dσ is a double-well potential. Let j = 1 when d = 1 and j = 2 when d = 2 or 3. We assume that g ∈ C^j+1(R) and satisfies the conditions g' (φ) ≥ -C₁, G(φ) ≥ -C₂and (φ - m(φ))g(φ) - C₃(m(φ))G(s) ≥ -C₄(m(φ)) (C₅(ρ) ≤ Cl(m(φ)) ≤ C6(%), l = 3, 4, whenever |m(φ)| ≤ ρ), where %, C₁, C₂, C₄≥ 0, C₃, C₅, C₆> 0 and m(φ) = 1 |Ω| ∫ _Ωφ(x)dx. For instance, g(φ) = Σ^2p-1_k=1akφ^k, p ∈ N, p ≥ 2, a_2p-1> 0, satisfies all the above-mentioned conditions. We then prove a well-posedness result, the existence of the global attractor and a family of exponential attractors in the phase space V_j= D(N^j/2) × D(N^j/2) equipped with the norm ||(ψ, φ)||V_j= (||N^j/2ψ||²+m(ψ)²+ ||N^j/2φ||²+m(φ)²)^1/2. Moreover, we demonstrate that the global attractor is upper semicontinuous at ϵ = 0 in the metric induced by the norm ||.||V_j+1. In addition, the exponential attractors are proven to be Hölder continuous at ϵ = 0 in the metric induced by the norm ||.||V_j. Our results improve a recent work by Bonfoh and Enyi [Comm. Pure Appl. Anal. 2016; 35:1077-1105] where the following additional growth condition |g''(φ)| ≤ C₇(|φ| p + 1), C₇> 0, p > 0 is arbitrary when d = 1, 2 and p ∈ [0, 3] when d = 3, was required, preventing g to be a polynomial of any arbitrary odd degree with a strictly positive leading coefficient in three space dimension.