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On the 3D Cahn-Hilliard equation with inertial term. (English) Zbl 1239.35160
Summary: We study the modified Cahn-Hilliard equation proposed by Galenko et al. in order to account for rapid spinodal decomposition in certain glasses. This equation contains, as additional term, the second-order time derivative of the (relative) concentration multiplied by a (small) positive coefficient $\epsilon$. Thus, in absence of viscosity effects, we are in presence of a Petrovsky type equation and the solutions do not regularize in finite time. Many results are known in one spatial dimension. However, even in two spatial dimensions, the problem of finding a unique solution satisfying given initial and boundary conditions is far from being trivial. A fairly complete analysis of the 2D case has been recently carried out by M. Grasselli, G. Schimperna and S. Zelik [Commun. Partial Differ. Equations 34, No. 2, 137–170 (2009; Zbl 1173.35086)]. The 3D case is still rather poorly understood but for the existence of energy bounded solutions. Taking advantage of this fact, A. Segatti [Math. Models Methods Appl. Sci. 17, No. 3, 411–437 (2007; Zbl 1131.35082)] has investigated the asymptotic behavior of a generalized dynamical system which can be associated with the equation. Here we take a step further by establishing the existence and uniqueness of a global weak solution, provided that $\epsilon$ is small enough. More precisely, we show that there exists ${\epsilon }_{0}>0$ such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of $\epsilon \in \left(0,{\epsilon }_{0}\right)$ which goes to $+\infty$ as $\epsilon$ tends to 0. This result allows us to construct a semigroup ${S}_{\epsilon }\left(t\right)$ on an appropriate (bounded) phase space and, besides, to prove the existence of a global attractor. Finally, we show a regularity result for the attractor by using a decomposition method and we discuss the existence of an exponential attractor.
##### MSC:
 35Q82 PDEs in connection with statistical mechanics 35B40 Asymptotic behavior of solutions of PDE 82C26 Dynamic and nonequilibrium phase transitions (general) 35B41 Attractors (PDE) 35D30 Weak solutions of PDE 37L30 Attractors and their dimensions, Lyapunov exponents