Some properties of solutions for the sixth-order Cahn-Hilliard-type equation. (English) Zbl 1258.35113

Summary: We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. We also discuss the existence of global attractor.


35K35 Initial-boundary value problems for higher-order parabolic equations
35K58 Semilinear parabolic equations
35B44 Blow-up in context of PDEs
35B41 Attractors
Full Text: DOI


[1] G. Gompper and J. Goos, “Fluctuating interfaces in microemulsion and sponge phases,” Physical Review E, vol. 50, no. 2, pp. 1325-1335, 1994.
[2] G. Gompper and M. Kraus, “Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations,” Physical Review E, vol. 47, no. 6, pp. 4289-4300, 1993.
[3] G. Gompper and M. Kraus, “Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations,” Physical Review E, vol. 47, no. 6, pp. 4301-4312, 1993.
[4] J. D. Evans, V. A. Galaktionov, and J. R. King, “Unstable sixth-order thin film equation. I. Blow-up similarity solutions,” Nonlinearity, vol. 20, no. 8, pp. 1799-1841, 2007. · Zbl 1173.35562
[5] J. D. Evans, V. A. Galaktionov, and J. R. King, “Unstable sixth-order thin film equation. II. Global similarity patterns,” Nonlinearity, vol. 20, no. 8, pp. 1843-1881, 2007. · Zbl 1173.35530
[6] Z. Li and C. Liu, “On the nonlinear instability of travelingwaves for a sixth order parabolic equation,” Abstract and Applied Analysis, vol. 2012, Article ID 739156, 17 pages, 2012. · Zbl 1257.35061
[7] C. Liu, “Qualitative properties for a sixth-order thin film equation,” Mathematical Modelling and Analysis, vol. 15, no. 4, pp. 457-471, 2010. · Zbl 1221.35101
[8] C. Liu and Y. Tian, “Weak solutions for a sixth-order thin film equation,” The Rocky Mountain Journal of Mathematics, vol. 41, no. 5, pp. 1547-1565, 2011. · Zbl 1227.35150
[9] I. Pawłow and W. M. Zaj\caczkowski, “A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,” Communications on Pure and Applied Analysis, vol. 10, no. 6, pp. 1823-1847, 2011. · Zbl 1229.35108
[10] G. Schimperna and I. Pawłow, “On a class of Cahn-Hilliard models with nonlinear diffusion,” http://arxiv.org/abs/1106.1581. · Zbl 1276.35101
[11] C. Liu, “Regularity of solutions for a sixth order nonlinear parabolic equation in two spacedimensions,” Annales Polonici Mathematici. In press.
[12] M. D. Korzec, P. L. Evans, A. Münch, and B. Wagner, “Stationary solutions of driven fourth- and sixth-order Cahn-Hilliard-type equations,” SIAM Journal on Applied Mathematics, vol. 69, no. 2, pp. 348-374, 2008. · Zbl 1171.34037
[13] B. Nicolaenko, B. Scheurer, and R. Temam, “Some global dynamical properties of a class of pattern formation equations,” Communications in Partial Differential Equations, vol. 14, no. 2, pp. 245-297, 1989. · Zbl 0691.35019
[14] T. Dłotko, “Global attractor for the Cahn-Hilliard equation in H2 and H3,” Journal of Differential Equations, vol. 113, no. 2, pp. 381-393, 1994. · Zbl 0828.35015
[15] D. Li and C. Zhong, “Global attractor for the Cahn-Hilliard system with fast growing nonlinearity,” Journal of Differential Equations, vol. 149, no. 2, pp. 191-210, 1998. · Zbl 0912.35029
[16] H. Wu and S. Zheng, “Global attractor for the 1-D thin film equation,” Asymptotic Analysis, vol. 51, no. 2, pp. 101-111, 2007. · Zbl 1223.35081
[17] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988. · Zbl 0662.35001
[18] A. Pazy, “Semigroups of linear operators and applications to partial differential equations,” in Applied Mathematical Sciences, vol. 44, Springer, 1983. · Zbl 0516.47023
[19] L. Song, Y. Zhang, and T. Ma, “Global attractor of the Cahn-Hilliard equation in Hk spaces,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 53-62, 2009. · Zbl 1173.35028
[20] L. Song, Y. Zhang, and T. Ma, “Global attractor of a modified Swift-Hohenberg equation in Hk spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 183-191, 2010. · Zbl 1180.35126
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.