-
Previous Article
Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition
- EECT Home
- This Issue
-
Next Article
An Ingham--Müntz type theorem and simultaneous observation problems
Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term
1. | Department of Mathematics, Jilin University, Changchun 130012, China |
2. | Department of Mathematics, and Key Laboratory of Symbolic Computation, and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012 |
References:
[1] |
S. J. Deng, W. K. Wang and H. L. Zhao, Existence theory and $L^p$ estimates for the solution of nonlinear viscous wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 4404-4414.
doi: 10.1016/j.nonrwa.2010.05.024. |
[2] |
L. Duan, S. Q. Liu and H. J. Zhao, A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation, J. Math. Anal. Appl., 372 (2010), 666-678.
doi: 10.1016/j.jmaa.2010.06.009. |
[3] |
P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A, 287 (2001), 190-197.
doi: 10.1016/S0375-9601(01)00489-3. |
[4] |
P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125.
doi: 10.1103/PhysRevE.71.046125. |
[5] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.
doi: 10.1103/PhysRevE.47.4289. |
[6] |
G. Gomppern and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.
doi: 10.1103/PhysRevE.47.4301. |
[7] |
G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335.
doi: 10.1103/PhysRevE.50.1325. |
[8] |
D. Jou, J. Casas-Vazquez and G. Lebon, Extended irreversible thermodynamics, Rep. Prog. Phys., 51 (1988), 1105-1179.
doi: 10.1088/0034-4885/51/8/002. |
[9] |
N. Y. Li and L. F. Mi, Pointwise estimates of solutions for the Cahn-Hilliard equation with inertial term in multi-dimensions, J. Math. Anal. Appl., 397 (2013), 75-87.
doi: 10.1016/j.jmaa.2012.07.040. |
[10] |
C. Liu and Z. Wang, Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 1087-1104.
doi: 10.3934/cpaa.2014.13.1087. |
[11] |
C. Liu and Z. Wang, Optimal control for a sixth order nonlinear parabolic equation, Mathematical Methods in the Applied Sciences, 38 (2015), 247-262.
doi: 10.1002/mma.3063. |
[12] |
C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions, Annales Polonici Mathematici, 107 (2013), 271-291.
doi: 10.4064/ap107-3-4. |
[13] |
I. Pawłow and W. Zajăczkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal., 10 (2011), 1823-1847.
doi: 10.3934/cpaa.2011.10.1823. |
[14] |
G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63.
doi: 10.1137/110835608. |
[15] |
W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241.
doi: 10.1016/j.jmaa.2009.12.013. |
[16] |
W. K. Wang and Z. G. Wu, Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions, J. Math. Anal. Appl., 387 (2012), 349-358.
doi: 10.1016/j.jmaa.2011.09.016. |
show all references
References:
[1] |
S. J. Deng, W. K. Wang and H. L. Zhao, Existence theory and $L^p$ estimates for the solution of nonlinear viscous wave equation, Nonlinear Anal. Real World Appl., 11 (2010), 4404-4414.
doi: 10.1016/j.nonrwa.2010.05.024. |
[2] |
L. Duan, S. Q. Liu and H. J. Zhao, A note on the optimal temporal decay estimates of solutions to the Cahn-Hilliard equation, J. Math. Anal. Appl., 372 (2010), 666-678.
doi: 10.1016/j.jmaa.2010.06.009. |
[3] |
P. Galenko, Phase-field model with relaxation of the diffusion flux in nonequilibrium solidification of a binary system, Phys. Lett. A, 287 (2001), 190-197.
doi: 10.1016/S0375-9601(01)00489-3. |
[4] |
P. Galenko and D. Jou, Diffuse-interface model for rapid phase transformations in nonequilibrium systems, Phys. Rev. E, 71 (2005), 046125.
doi: 10.1103/PhysRevE.71.046125. |
[5] |
G. Gompper and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. I. Gaussian interface fluctuations, Phys. Rev. E, 47 (1993), 4289-4300.
doi: 10.1103/PhysRevE.47.4289. |
[6] |
G. Gomppern and M. Kraus, Ginzburg-Landau theory of ternary amphiphilic systems. II. Monte Carlo simulations, Phys. Rev. E, 47 (1993), 4301-4312.
doi: 10.1103/PhysRevE.47.4301. |
[7] |
G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E, 50 (1994), 1325-1335.
doi: 10.1103/PhysRevE.50.1325. |
[8] |
D. Jou, J. Casas-Vazquez and G. Lebon, Extended irreversible thermodynamics, Rep. Prog. Phys., 51 (1988), 1105-1179.
doi: 10.1088/0034-4885/51/8/002. |
[9] |
N. Y. Li and L. F. Mi, Pointwise estimates of solutions for the Cahn-Hilliard equation with inertial term in multi-dimensions, J. Math. Anal. Appl., 397 (2013), 75-87.
doi: 10.1016/j.jmaa.2012.07.040. |
[10] |
C. Liu and Z. Wang, Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions, Commun. Pure Appl. Anal., 13 (2014), 1087-1104.
doi: 10.3934/cpaa.2014.13.1087. |
[11] |
C. Liu and Z. Wang, Optimal control for a sixth order nonlinear parabolic equation, Mathematical Methods in the Applied Sciences, 38 (2015), 247-262.
doi: 10.1002/mma.3063. |
[12] |
C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions, Annales Polonici Mathematici, 107 (2013), 271-291.
doi: 10.4064/ap107-3-4. |
[13] |
I. Pawłow and W. Zajăczkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal., 10 (2011), 1823-1847.
doi: 10.3934/cpaa.2011.10.1823. |
[14] |
G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal., 45 (2013), 31-63.
doi: 10.1137/110835608. |
[15] |
W. K. Wang and W. J. Wang, The pointwise estimates of solutions for semilinear dissipative wave equation in multi-dimensions, J. Math. Anal. Appl., 366 (2010), 226-241.
doi: 10.1016/j.jmaa.2009.12.013. |
[16] |
W. K. Wang and Z. G. Wu, Optimal decay rate of solutions for Cahn-Hilliard equation with inertial term in multi-dimensions, J. Math. Anal. Appl., 387 (2012), 349-358.
doi: 10.1016/j.jmaa.2011.09.016. |
[1] |
Irena Pawłow, Wojciech M. Zajączkowski. On a class of sixth order viscous Cahn-Hilliard type equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 517-546. doi: 10.3934/dcdss.2013.6.517 |
[2] |
Irena Pawłow, Wojciech M. Zajączkowski. A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1823-1847. doi: 10.3934/cpaa.2011.10.1823 |
[3] |
Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859 |
[4] |
Cecilia Cavaterra, Maurizio Grasselli, Hao Wu. Non-isothermal viscous Cahn-Hilliard equation with inertial term and dynamic boundary conditions. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1855-1890. doi: 10.3934/cpaa.2014.13.1855 |
[5] |
Maurizio Grasselli, Nicolas Lecoq, Morgan Pierre. A long-time stable fully discrete approximation of the Cahn-Hilliard equation with inertial term. Conference Publications, 2011, 2011 (Special) : 543-552. doi: 10.3934/proc.2011.2011.543 |
[6] |
Alain Miranville. Existence of solutions for Cahn-Hilliard type equations. Conference Publications, 2003, 2003 (Special) : 630-637. doi: 10.3934/proc.2003.2003.630 |
[7] |
Nguyen Huy Tuan. Existence and limit problem for fractional fourth order subdiffusion equation and Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4551-4574. doi: 10.3934/dcdss.2021113 |
[8] |
Georgia Karali, Yuko Nagase. On the existence of solution for a Cahn-Hilliard/Allen-Cahn equation. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 127-137. doi: 10.3934/dcdss.2014.7.127 |
[9] |
Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31 |
[10] |
Anna Kostianko, Sergey Zelik. Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions. Communications on Pure and Applied Analysis, 2015, 14 (5) : 2069-2094. doi: 10.3934/cpaa.2015.14.2069 |
[11] |
Cristina Pocci. On singular limit of a nonlinear $p$-order equation related to Cahn-Hilliard and Allen-Cahn evolutions. Evolution Equations and Control Theory, 2013, 2 (3) : 517-530. doi: 10.3934/eect.2013.2.517 |
[12] |
Changchun Liu, Hui Tang. Existence of periodic solution for a Cahn-Hilliard/Allen-Cahn equation in two space dimensions. Evolution Equations and Control Theory, 2017, 6 (2) : 219-237. doi: 10.3934/eect.2017012 |
[13] |
Ahmad Makki, Alain Miranville. Existence of solutions for anisotropic Cahn-Hilliard and Allen-Cahn systems in higher space dimensions. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 759-775. doi: 10.3934/dcdss.2016027 |
[14] |
Makoto Okumura, Takeshi Fukao, Daisuke Furihata, Shuji Yoshikawa. A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition. Communications on Pure and Applied Analysis, 2022, 21 (2) : 355-392. doi: 10.3934/cpaa.2021181 |
[15] |
L. Chupin. Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation. Discrete and Continuous Dynamical Systems - B, 2003, 3 (1) : 45-68. doi: 10.3934/dcdsb.2003.3.45 |
[16] |
Andrea Signori. Optimal treatment for a phase field system of Cahn-Hilliard type modeling tumor growth by asymptotic scheme. Mathematical Control and Related Fields, 2020, 10 (2) : 305-331. doi: 10.3934/mcrf.2019040 |
[17] |
Ciprian G. Gal, Maurizio Grasselli. Longtime behavior of nonlocal Cahn-Hilliard equations. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 145-179. doi: 10.3934/dcds.2014.34.145 |
[18] |
Desheng Li, Xuewei Ju. On dynamical behavior of viscous Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2207-2221. doi: 10.3934/dcds.2012.32.2207 |
[19] |
Laurence Cherfils, Alain Miranville, Sergey Zelik. On a generalized Cahn-Hilliard equation with biological applications. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2013-2026. doi: 10.3934/dcdsb.2014.19.2013 |
[20] |
Álvaro Hernández, Michał Kowalczyk. Rotationally symmetric solutions to the Cahn-Hilliard equation. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 801-827. doi: 10.3934/dcds.2017033 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]