# American Institute of Mathematical Sciences

April  2013, 6(2): 517-546. doi: 10.3934/dcdss.2013.6.517

## On a class of sixth order viscous Cahn-Hilliard type equations

 1 System Research Institute, Polish Academy of Sciences, Newelska 6, 01-447 Warsaw 2 Institute of Mathematics, Polish Academy of Sciences, Śniadeckich 8, 00-950 Warsaw

Received  July 2011 Revised  February 2012 Published  November 2012

An initial-boundary-value problem for a class of sixth order viscous Cahn-Hilliard type equations with a nonlinear diffusion is considered. The study is motivated by phase-field modelling of various spatial structures, for example arising in oil-water-surfactant mixtures and in modelling of crystal growth on atomic length, known as phase field crystal model. For such problem we prove the existence and uniqueness of a global in time regular solution. First the finite-time existence is proved by means of the Leray-Schauder fixed point theorem. Then, due to suitable estimates, the finite-time solution is extended step by step on the infinite time interval.
Citation: Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517
##### References:

show all references

##### References:
 [1] Irena Pawłow, Wojciech M. Zajączkowski. A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1823-1847. doi: 10.3934/cpaa.2011.10.1823 [2] Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859 [3] Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations & Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315 [4] Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 [5] Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207 [6] Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 [7] Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021113 [8] Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 [9] Kelong Cheng, Cheng Wang, Steven M. Wise, Zixia Yuan. Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2211-2229. doi: 10.3934/dcdss.2020186 [10] Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 [11] Irena Pawłow, Wojciech M. Zajączkowski. Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids. Conference Publications, 2007, 2007 (Special) : 824-833. doi: 10.3934/proc.2007.2007.824 [12] Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855 [13] Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations & Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517 [14] Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations & Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 [15] Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 [16] Riccarda Rossi. On two classes of generalized viscous Cahn-Hilliard equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 405-430. doi: 10.3934/cpaa.2005.4.405 [17] Pablo Álvarez-Caudevilla. Existence and multiplicity of stationary solutions for a Cahn--Hilliard-type equation in $\mathbb{R}^N$. Conference Publications, 2015, 2015 (special) : 10-18. doi: 10.3934/proc.2015.0010 [18] Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013 [19] Sergey Zelik, Jon Pennant. Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2013, 12 (1) : 461-480. doi: 10.3934/cpaa.2013.12.461 [20] Satoshi Kosugi, Yoshihisa Morita, Shoji Yotsutani. Stationary solutions to the one-dimensional Cahn-Hilliard equation: Proof by the complete elliptic integrals. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 609-629. doi: 10.3934/dcds.2007.19.609

2020 Impact Factor: 2.425