Abstract and Applied Analysis

Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

Abstract

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.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 414590, 24 pages.

Dates
First available in Project Euclid: 28 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1364475888

Digital Object Identifier
doi:10.1155/2012/414590

Mathematical Reviews number (MathSciNet)
MR2991005

Zentralblatt MATH identifier
1258.35113

Citation

Wang, Zhao; Liu, Changchun. Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation. Abstr. Appl. Anal. 2012 (2012), Article ID 414590, 24 pages. doi:10.1155/2012/414590. https://projecteuclid.org/euclid.aaa/1364475888

References

• G. Gompper and J. Goos, “Fluctuating interfaces in microemulsion and sponge phases,” Physical Review E, vol. 50, no. 2, pp. 1325–1335, 1994.
• 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.
• 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.
• 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.
• 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.
• 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.
• C. Liu, “Qualitative properties for a sixth-order thin film equation,” Mathematical Modelling and Analysis, vol. 15, no. 4, pp. 457–471, 2010.
• 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.
• I. Pawłow and W. M. Zajączkowski, “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.
• G. Schimperna and I. Pawłow, “On a class of Cahn-Hilliard čommentComment on ref. [15?]: Please update the information of this reference, if possible.models with nonlinear diffusion,” http://arxiv.org/abs/1106.1581.
• C. Liu, “Regularity of solutions for a sixth order nonlinear parabolic equation in two spacečommentComment on ref. [12?]: Please update the information of this reference, if possible.dimensions,” Annales Polonici Mathematici. In press.
• 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.
• 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.
• T. Dłotko, “Global attractor for the Cahn-Hilliard equation in ${H}^{2}$ and ${H}^{3}$,” Journal of Differential Equations, vol. 113, no. 2, pp. 381–393, 1994.
• 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.
• H. Wu and S. Zheng, “Global attractor for the 1-D thin film equation,” Asymptotic Analysis, vol. 51, no. 2, pp. 101–111, 2007.
• R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988.
• A. Pazy, “Semigroups of linear operators and applications to partial differential equations,” in Applied Mathematical Sciences, vol. 44, Springer, 1983.
• L. Song, Y. Zhang, and T. Ma, “Global attractor of the Cahn-Hilliard equation in ${H}^{k}$ spaces,” Journal of Mathematical Analysis and Applications, vol. 355, no. 1, pp. 53–62, 2009.
• L. Song, Y. Zhang, and T. Ma, “Global attractor of a modified Swift-Hohenberg equation in ${H}^{k}$ spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 1, pp. 183–191, 2010.