We consider a conserved phase-field system on a tridimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ϑ. These effects are represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore the system consists of a linear integrodifferential equation for ϑ which is coupled with a viscous Cahn-Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity φ and the viscosity effects are taken into account by the term −α∆χt, for some α ≥ 0. Thus, we formulate a Cauchy-Neumann problem depending on α. Assuming suitable conditions on k, we prove that this problem generates a dissipative strongly continuous semigroup S α (t) on an appropriate phase space accounting for the past histories of ϑ as well as for the conservation of the spatial means of the enthalpy ϑ + χ and of the order parameter. We first show, for any α ≥ 0, the existence of the global attractor Aα. Also, in the viscous case (α > 0), we prove the finiteness of the fractal dimension and the smoothness of Aα.

We deal with the memory relaxation of the viscous Cahn-Hilliard equation in 3-D, covering the wellknown hyperbolic version of the model. We study the long-term dynamic of the system in dependence of the scaling parameter of the memory kernel ε and of the viscosity coefficient . In particular we construct a family of exponential attractors, which is robust as both ε and go to zero, provided that ε is linearly controlled by .where > 0 is chosen as in (6). Arguing exactly as in Lemmas 4.5 and 4.7, keeping in mind that the initial conditions are null, it is immediate to realize thatThis can be done arguing as in the proof of [9, Lemma 5.5] (see also [8]) by exploiting the exponential decay of ε , the straightforward inequalities M 0 ε can conclude that the global attractor A ε, has finite fractal dimension, which is uniform with respect to ε and . In addition, arguing as in [9, Section 7] with obvious changes (see also [20]), it is possible to prove that the global attractor is upper semicontinuous as (ε, ) → (0, 0), namely lim (ε, )→(0,0) dist(A ε, , A 0,0 ) = 0

Let H be a real separable Hilbert space and A : D(A) → H be a positive and self-adjoint (unbounded) operator. We consider the identification problem consisting in searching for an H-valued function u and a couple of real numbers λ and μ, the first one being positive, that fulfil the initial-value problem and the additional constraintsA r/2 u(T ) 2 = ϕ and A s/2 u(T ) 2 = ψ, where we denote by A s and A r the powers of A with exponents r < s. Provided that the given data u 0 ∈ H, u 0 and ϕ, ψ > 0 satisfy proper a priori limitations, by means of a finite-dimensional approximation scheme, we construct a unique solution (u, λ, μ) on the whole interval [0, T ], and exhibit an explicit continuous dependence estimate of Lipschitz type with respect to the data. Also, we provide specific applications to second-and fourth-order parabolic initial-boundary-value problems.

Let be a complex Hilbert space and and be nonnegative and selfadjoint operators. We study the inverse problem consisting in the identification of the function and two constants α, (diffusion coefficients) that fulfill the initial-value problem and the additional conditions Under suitable assumptions on the operators A and B, and on the data and , we shall construct a solution and prove its uniqueness and continuous dependence on the data. Applications are considered.

Let H be a real separable Hilbert space and A : D(A) → H be a positive and self-adjoint (unbounded) operator, and denote by A σ its power of exponent σ ∈ [−1, 1). We consider the identification problem consisting in searching for a function u : [0, T ] → H and a real constant µ that fulfill the initial-value problemand the additional conditionwhere u 0 ∈ H, u 0 = 0 and α, β ≥ 0, α + β > 0 and ρ > 0 are given. By means of a finite-dimensional approximation scheme, we construct a unique solution (u, µ) of suitable regularity on the whole interval [0, T ], and exhibit an explicit continuous dependence estimate of Lipschitz-type with respect to the data u 0 and ρ. Also, we provide specific applications to second and fourth-order parabolic initial-boundary value problems.