May  2014, 13(3): 1087-1104. doi: 10.3934/cpaa.2014.13.1087

Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions

1. 

Department of Mathematics, Jilin University, Changchun 130012, China

Received  April 2013 Revised  October 2013 Published  December 2013

In this paper, we study the time periodic solution of a sixth order nonlinear parabolic equation, which arises in oil-water-surfactant mixtures. Based on Leray-Schauder's fixed point theorem and Campanato spaces, we prove the existence of time-periodic solutions in two space dimensions.
Citation: Changchun Liu, Zhao Wang. Time periodic solutions for a sixth order nonlinear parabolic equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1087-1104. doi: 10.3934/cpaa.2014.13.1087
References:
[1]

Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation,, \emph{J. Math. Anal. Appl.}, 313 (2006), 311.  doi: 10.1016/j.jmaa.2005.08.073.  Google Scholar

[2]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems,, \emph{Math. Z.}, 179 (1982), 437.  doi: 10.1007/BF01215058.  Google Scholar

[3]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases,, \emph{Phys. Rev. E}, 50 (1994), 1325.   Google Scholar

[4]

C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions,, \emph{Annales Polonici Mathematici}, 107 (2013), 271.  doi: 10.4064/ap107-3-4.  Google Scholar

[5]

I. Pawłow and W. Zajączkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1823.  doi: 10.3934/cpaa.2011.10.1823.  Google Scholar

[6]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, \emph{SIAM J. Math. Anal.}, 45 (2013), 31.  doi: 10.1137/110835608.  Google Scholar

[7]

R. Wang, The Schauder theory of the boundary value problem for parabolic problem equations,, \emph{Acta Sci. Nature Univ. Jilin.}, 2 (1964), 35.   Google Scholar

[8]

Y. Wang and Y. Zhang, Time-periodic solutions to a nonlinear parabolic type equation of higher order,, \emph{Acta Math. Appl. Sin., 24 (2008), 129.  doi: 10.1007/s10255-006-6174-3.  Google Scholar

[9]

L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard type equation,, \emph{Mathematical and Computer Modelling}, 48 (2008), 11.  doi: 10.1016/j.mcm.2007.09.001.  Google Scholar

[10]

J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentials and sources,, \emph{Appl. Math. Comput.}, 211 (2009), 211.  doi: 10.1016/j.amc.2009.01.038.  Google Scholar

show all references

References:
[1]

Y. Fu and B. Guo, Time periodic solution of the viscous Camassa-Holm equation,, \emph{J. Math. Anal. Appl.}, 313 (2006), 311.  doi: 10.1016/j.jmaa.2005.08.073.  Google Scholar

[2]

M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems,, \emph{Math. Z.}, 179 (1982), 437.  doi: 10.1007/BF01215058.  Google Scholar

[3]

G. Gompper and J. Goos, Fluctuating interfaces in microemulsion and sponge phases,, \emph{Phys. Rev. E}, 50 (1994), 1325.   Google Scholar

[4]

C. Liu, Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions,, \emph{Annales Polonici Mathematici}, 107 (2013), 271.  doi: 10.4064/ap107-3-4.  Google Scholar

[5]

I. Pawłow and W. Zajączkowski, A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,, \emph{Commun. Pure Appl. Anal.}, 10 (2011), 1823.  doi: 10.3934/cpaa.2011.10.1823.  Google Scholar

[6]

G. Schimperna and I. Pawłow, On a class of Cahn-Hilliard models with nonlinear diffusion,, \emph{SIAM J. Math. Anal.}, 45 (2013), 31.  doi: 10.1137/110835608.  Google Scholar

[7]

R. Wang, The Schauder theory of the boundary value problem for parabolic problem equations,, \emph{Acta Sci. Nature Univ. Jilin.}, 2 (1964), 35.   Google Scholar

[8]

Y. Wang and Y. Zhang, Time-periodic solutions to a nonlinear parabolic type equation of higher order,, \emph{Acta Math. Appl. Sin., 24 (2008), 129.  doi: 10.1007/s10255-006-6174-3.  Google Scholar

[9]

L. Yin, Y. Li, R. Huang and J. Yin, Time periodic solutions for a Cahn-Hilliard type equation,, \emph{Mathematical and Computer Modelling}, 48 (2008), 11.  doi: 10.1016/j.mcm.2007.09.001.  Google Scholar

[10]

J. Yin, Y. Li and R. Huang, The Cahn-Hilliard type equations with periodic potentials and sources,, \emph{Appl. Math. Comput.}, 211 (2009), 211.  doi: 10.1016/j.amc.2009.01.038.  Google Scholar

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