SUFFICIENT CONDITIONS FOR THE CONTINUITY OF INERTIAL MANIFOLDS FOR SINGULARLY PERTURBED PROBLEMS

We consider a nonlinear evolution equation in the form together with its singular limit problem as ε → 0 U_t + A_εU + N_εG_ε(U) = 0, (E_ε) U_t + AU + NG(U) = 0, where ε ∈ (0, 1] (possibly ε = 0), A_ε and A are linear closed (possibly) unbounded operators, N_ε and N are linear (possibly) unbounded operators, G_ε and G are nonlinear functions. We give sufficient conditions on A_ε, N_ε and G_ε (and also A, N and G) that guarantee not only the existence of an inertial manifold of dimension independent of ε for (E_ε) on a Banach space H, but also the Hölder continuity, lower and upper semicontinuity at ε = 0 of the intersection of the inertial manifold with a bounded absorbing set. Applications to higher order viscous Cahn-Hilliard-Oono equations, the hyperbolic type equations and the phase-field systems, subject to either Neumann or Dirichlet boundary conditions (BC) (in which case Ω ⊂ ℝ^d is a bounded domain with smooth bound-ary) or periodic BC (in which case Ω = Π^di=1^{(0, L}i), L_i > 0), d = 1, 2 or 3, are considered. These three classes of dissipative equations read (E) ϕ_t + N(εϕ_t + N^α+1 ϕ + Nϕ + g(ϕ)) + σϕ = 0, α ∈ ℕ, (P_ε) εϕ_tt + ϕ_t + N^α (Nϕ + g(ϕ)) + σϕ = 0, α = 0, 1, (H_ε) and { ϕt + N^α (Nϕ + g(ϕ) − u) + σϕ = 0, εu_t + ϕ_t + Nu = 0, respectively, where σ ≥ 0 and the Laplace operator is defined as α = 0, 1, (S_ε) N = −∆: D(N) = {ψ ∈ H² (Ω), ψ subject to the BC} →^˙L² (Ω) or L² (Ω). We assume that, for a given real number c₁ > 0, there exists a positive integer n = n(c₁) such that λ_n+1 −λ_n > c₁, where {λ_k }_k∈ℕ∗ are the eigenvalues of N. There exists a real number M > 0 such that the nonlinear function g: V_j → V_j satisfies the conditions ‖g(ψ)‖_j ≤ M and ‖g(ψ) − g(φ)‖_Vj ≤ M ‖ψ − φ‖_Vj, ∀ψ, φ ∈ V_j, where V_j = D(N^j/2), j = 1 for Problems (P_ε) and (S_ε) and j = 0, 2α for Problem (H_ε). We further require g ∈ C¹ (V_j, V_j), ‖g^′ (ψ)φ‖_j ≤ M ‖φ‖_j for Problems (H_ε) and (S_ε).