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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 ε. 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 ε is small enough. More precisely, we show that there exists ε 0 >0 such that well-posedness holds if (suitable) norms of the initial data are bounded by a positive function of ε(0,ε 0 ) which goes to + as ε tends to 0. This result allows us to construct a semigroup S ε (t) 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:
35Q82PDEs in connection with statistical mechanics
35B40Asymptotic behavior of solutions of PDE
82C26Dynamic and nonequilibrium phase transitions (general)
35B41Attractors (PDE)
35D30Weak solutions of PDE
37L30Attractors and their dimensions, Lyapunov exponents
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