Abstract
We consider a non-isothermal modified Cahn--Hilliard equation which was previously analyzed by M. Grasselli et al. Such an equation is characterized by an inertial term and a viscous term and it is coupled with a hyperbolic heat equation. The resulting system was studied in the case of no-flux boundary conditions. Here we analyze the case in which the order parameter is subject to a dynamic boundary condition. This assumption requires a more refined strategy to extend the previous results to the present case. More precisely, we first prove the well-posedness for solutions with bounded energy as well as for weak solutions. Then we establish the existence of a global attractor. Finally, we prove the convergence of any given weak solution to a single equilibrium by using a suitable Lojasiewicz--Simon inequality.
Highlights
The Cahn–Hilliard equation is a cornerstone in Materials Science since it gives a fairly good description of phase separation processes in binary alloys
Α > 0 is a viscosity parameter accounting for possible presence of microforces and f is the derivative of a given double-well potential
We recall that the classical Cahn–Hilliard equation corresponds to the case ε = α = 0
Summary
The Cahn–Hilliard equation is a cornerstone in Materials Science since it gives a fairly good description of phase separation processes in binary alloys (see, e.g., [7, 40, 41] and references therein). A modification of the Cahn–Hilliard equation has been proposed in [18] to account for rapid spinodal decomposition in certain materials (see [19, 20]) This modified equation reads as follows εχtt + χt − ∆μ = 0, where ε > 0 is a relaxation time, χ represents the (relative) concentration of one component and μ is the so-called chemical potential given by μ = −∆χ + αχt + f (χ). In the present case a regularizing effect for χ is missing due to the presence of the dynamic boundary condition (1.6) This entails that the equation (1.3) must be understood in a more generalized way with respect to [27] (see Remark 2.4 below). Among the open issues it is worth mentioning the existence of a family of exponential attractors and its robustness with respect to σ, ε and α (see [22] for the isothermal case)
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