# zbMATH — the first resource for mathematics

Dynamics of the viscous Cahn-Hilliard equation. (English) Zbl 1151.35008
The viscous Cahn-Hilliard problem of the form: $\begin{gathered} (1-\nu) u_t= -\Delta(\Delta u+ f(u)- \nu u_t)\quad\text{in }\Omega,\\ u(t,x)=\Delta u(t, x)= 0\quad\text{in }\partial\Omega,\\ u(0,x)= u_0(x)\end{gathered}\tag{1.1}$ is considered. There is assumed, that $$\nu\in[0, 1]$$, $$f\in C^1(\mathbb{R},\mathbb{R})$$ satisfies suitable growth and dissipation conditions and $$\Omega$$ is a bounded smooth domain in $$\mathbb{R}^n$$, $$n\geq 3$$.
The aim of the paper is to study the nonlinear semigroup generated by (1.1) on the phase space $$H^1_0(\Omega)$$ (and on $$W^{1,p}_0(\Omega)$$ in case of local existence and regularity of solutions) with the equation satisfied in $$H^{-1}(\Omega)$$ when $$\nu> 0$$ or in $$H^{-3}(\Omega)$$ when $$\nu= 0$$. The main result states that the asymptotic dynamics of (1.1) is the same in a suitable sense for all values of $$\nu$$ in $$[0,1]$$; the global attractors $${\mathcal A}_\nu$$ are shown to be continuous for $$\nu\in[0, 1]$$. The asymptotic dynamics of the semilinear heat equation $$(\nu= 1)$$ is thus the same as the asymptotic dynamics of the Cahn-Hillard equation $$(\nu= 0)$$. Also the Hausdorff dimension of the global attractors $${\mathcal A}_\nu$$ for (1.1) is shown to be independent from $$\nu$$.

##### MSC:
 35B41 Attractors 37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations 35Q72 Other PDE from mechanics (MSC2000)
Full Text:
##### References:
 [1] Amann, H., Linear and quasilinear parabolic problems, (1995), Birkhäuser Basel [2] Arrieta, J.M.; Carvalho, A.N., Abstract parabolic problems with critical nonlinearities and applications to navier – stokes and heat equations, Trans. amer. math. soc., 352, 285-310, (1999) · Zbl 0940.35119 [3] Arrieta, J.M.; Carvalho, A.N., Spectral convergence and nonlinear dynamics of reaction – diffusion equations under perturbations of the domain, J. differential equations, 199, 143-178, (2004) · Zbl 1058.35028 [4] J.M. Arrieta, A.N. Carvalho, G. Lozada-Cruz, Dynamics in dumbbell domains II. The limiting problem, submitted for publication · Zbl 1172.35033 [5] Babin, A.; Vishik, M., Attractors of evolution equations, (1992), North-Holland Amsterdam · Zbl 0778.58002 [6] Ball, J.M., Global attractors for damped semilinear wave equations, Discrete contin. dyn. syst., 10, 31-52, (2004) · Zbl 1056.37084 [7] Brunovský, P.; Poláčik, P., The morse – smale structure of a generic reaction – diffusion equation in higher space dimension, J. differential equations, 135, 129-181, (1997) · Zbl 0868.35062 [8] Bruschi, S.M.; Carvalho, A.N.; Cholewa, J.W.; Dlotko, T., Uniform exponential dichotomy and continuity of attractors for singularly perturbed damped wave equation, J. dynam. differential equations, 18, 767-814, (2006) · Zbl 1103.35020 [9] Carvalho, A.N.; Cholewa, J.W., Continuation and asymptotics of solutions to semilinear parabolic equations with critical nonlinearities, J. math. anal. appl., 310, 557-578, (2005) · Zbl 1077.35031 [10] Carvalho, A.N.; Cholewa, J.W.; Dlotko, T., Strongly damped wave problems: bootstrapping and regularity of solutions, J. differential equations, 244, 9, 2310-2333, (2008) · Zbl 1151.35056 [11] Carvalho, A.N.; Langa, J.A.; Robinson, J.C.; Suarez, A., Characterization of non-autonomous attractors of a perturbed infinite-dimensional gradient system, J. differential equations, 236, 570-603, (2007) · Zbl 1119.37023 [12] Dlotko, T., Global attractor for the cahn – hilliard equation in $$H^2$$ and $$H^3$$, J. differential equations, 113, 381-393, (1994) · Zbl 0828.35015 [13] Elliott, C.M.; Stuart, A.M., Viscous cahn – hilliard equation II. analysis, J. differential equations, 128, 387-414, (1996) · Zbl 0855.35067 [14] Gatti, S.; Grasselli, M.; Miranville, A.; Pata, V., Hyperbolic relaxation of the viscous cahn – hilliard equation in 3-D, Math. models methods appl. sci., 15, 165-198, (2005) · Zbl 1073.35043 [15] Grinfeld, M.; Novick-Cohen, A., The viscous cahn – hilliard equation: Morse decomposition and structure of the global attractor, Trans. amer. math. soc., 351, 2375-2406, (1999) · Zbl 0927.35045 [16] Hale, J.K., Asymptotic behavior of dissipative systems, (1988), Amer. Math. Soc. Providence, RI · Zbl 0642.58013 [17] Hale, J.K.; Raugel, G., Lower semicontinuity of attractors of gradient systems and applications, Ann. mat. pura appl., 154, 281-326, (1989) · Zbl 0712.47053 [18] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer-Verlag Berlin · Zbl 0456.35001 [19] Kania, M.B., Global attractor for the perturbed viscous cahn – hilliard equation, Colloq. math., 109, 217-229, (2007) · Zbl 1116.35026 [20] Maier-Paape, S.; Mischaikow, K.; Wanner, T., Structure of the attractor of the cahn – hilliard equation on a square, Internat. J. bifur. chaos appl. sci. engrg., 17, 4, 1221-1263, (2007) · Zbl 1148.35008 [21] Novick-Cohen, A., On the viscous cahn – hilliard equation, (), 329-342 · Zbl 0632.76119 [22] Rossi, R., Global attractor for the weak solutions of a class of viscous cahn – hilliard equations, (), 247-268 · Zbl 1102.35024 [23] Temam, R., Infinite-dimensional dynamical systems in mechanics and physics, (1997), Springer-Verlag New York · Zbl 0871.35001 [24] Zheng, S.; Milani, A., Global attractors for singular perturbations of the cahn – hilliard equations, J. differential equations, 209, 101-139, (2005) · Zbl 1063.35041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.