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September  2015, 4(3): 315-324. doi: 10.3934/eect.2015.4.315

Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term

Aibo Liu 1, and  Changchun Liu 2,

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

Received  October 2014 Revised  March 2015 Published  September 2015

In this paper, we consider the Cauchy problem of a sixth order Cahn-Hilliard equation with the inertial term, \begin{eqnarray*} ku_{t t} + u_t - \Delta^3 u - \Delta(-a(u) \Delta u -\frac{a'(u)}2|\nabla u|^2 + f(u))=0. \end{eqnarray*} Based on Green's function method together with energy estimates, we get the global existence and optimal decay rate of solutions.
Keywords: inertial term., Sixth order Cahn-Hilliard, existence, optimal decay rate.
Mathematics Subject Classification: Primary: 35L30, 35L77; Secondary: 35B4.
Citation: Aibo Liu, Changchun Liu. Cauchy problem for a sixth order Cahn-Hilliard type equation with inertial term. Evolution Equations and Control Theory, 2015, 4 (3) : 315-324. doi: 10.3934/eect.2015.4.315
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.

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