Home Time periodic solutions for a sixth order nonlinear parabolic equation
Article
Licensed
Unlicensed Requires Authentication

Time periodic solutions for a sixth order nonlinear parabolic equation

  • Changchun Liu EMAIL logo and Zhao Wang EMAIL logo
Published/Copyright: April 29, 2016
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 1

Abstract

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, we prove the existence of time-periodic solutions.

Keywords: sixth order parabolic; time-periodic solution; existence
MSC: Primary 35B10; 35K35; Secondary 35K55

This work is supported by the National Science Foundation of China (no. 11471164).

  1. This paper has been communicated by Giuseppe Di Fazio

References

[1] Fu, Y.—Guo, B.: Time periodic solution of the viscous Camassa-Holm equation, J. Math. Anal. Appl. 313 (2006), 311–321.10.1016/j.jmaa.2005.08.073Search in Google Scholar

[2] Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nolinear Elliptic Systems, Princeton Univ. Press, Princeton, NJ, 1983.10.1515/9781400881628Search in Google Scholar

[3] Gilbarg, D.—Trudinger, N.: Elliptic partial differential equation of second order (2nd ed.), Springer-Verlag, Berlin, 1983.Search in Google Scholar

[4] Gompper, G.—Goos, J.: Fluctuating interfaces in microemulsion and sponge phases, Phys. Rev. E (3) 50 (1994), 1325–1335.10.1103/PhysRevE.50.1325Search in Google Scholar PubMed

[5] Liu, C.—Liu, A.—Tang, H.: Time periodic solutions for a driven sixth order Cahn-Hilliard type equation, Bound. Value Probl. (2013), 2013:73.10.1186/1687-2770-2013-73Search in Google Scholar

[6] Liu, C.: Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions, Ann. Polon. Math. 107 (2013), 271–291.10.4064/ap107-3-4Search in Google Scholar

[7] Pawłow, I.—Zaja̧czkowski, W.: A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures, Commun. Pure Appl. Anal. 10 (2011), 1823–1847.10.3934/cpaa.2011.10.1823Search in Google Scholar

[8] Schimperna, G.—Pawłow, I.: On a class of Cahn-Hilliard models with nonlinear diffusion, SIAM J. Math. Anal. 45 (2013), 31–63.10.1137/110835608Search in Google Scholar

[9] Wang, Y. P.—Zhang, Y. L.: Time-periodic solution to a nonlinear parabolic type equation of higher order, Acta Math. Appl. Sin. Engl. Ser. 24 (2008), 129–140.10.1007/s10255-006-6174-3Search in Google Scholar

[10] Wang, Z.—Liu, C.: Some properties of solutions for the sixth order Cahn-Hilliard type equation, Abstr. Appl. Anal. 2012 (2012), Article ID 41459, 24 pp., doi: 10.1155/2012/414590.10.1155/2012/414590Search in Google Scholar

[11] Yin, L.—Li, Y.—Huang, R.—Yin, J.: Time periodic solutions for a Cahn-Hilliard type equation, Math. Comput. Modelling 48 (2008), 11–18.10.1016/j.mcm.2007.09.001Search in Google Scholar

[12] Yin, J.: On the Cahn-Hilliard equation with nonlinear principal part, J. Partial Differ. Equ. 7 (1994), 77–96.Search in Google Scholar

Received: 2012-5-7
Accepted: 2013-4-22
Published Online: 2016-4-29
Published in Print: 2016-2-1

© 2016 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Research Article
  2. Prof. RNDr. Ján Jakubík, DrSc. passed away
  3. Research Article
  4. On the number of slim, semimodular lattices
  5. Research Article
  6. Quantum computational algebra with a non-commutative generalization
  7. Research Article
  8. Lexico groups and direct products of lattice ordered groups
  9. Research Article
  10. Values of limit periodic sequences and functions
  11. Research Article
  12. On representation of quasi-Hopf group coalgebras
  13. Research Article
  14. On the q-Bernstein polynomials of the logarithmic function in the case q > 1
  15. Research Article
  16. On some modification of Darboux property
  17. Research Article
  18. Separating sets by peripherally continuous functions
  19. Research Article
  20. Persistence and extinction of a stochastic delay predator-prey model in a polluted environment
  21. Research Article
  22. On O’Malley preponderantly continuous functions
  23. Research Article
  24. Inequalities for the normalized Jensen functional with applications
  25. Research Article
  26. Majorization of starlike and convex functions of complex order involving linear operators
  27. Research Article
  28. Fekete–Szegö problem for some starlike functions related to shell-like curves
  29. Research Article
  30. Central functions for classes of concave univalent functions
  31. Research Article
  32. Inclusion properties for classes of analytic functions associated with conic domains
  33. Research Article
  34. Uniqueness of meromorphic functions and nonlinear differential polynomials sharing a nonzero polynomial
  35. Research Article
  36. Solutions and constrained null-controllability for a differential-difference equation
  37. Research Article
  38. Oscillation criteria for quasi-linear neutral delay dynamic equations on time scale
  39. Research Article
  40. Time periodic solutions for a sixth order nonlinear parabolic equation
  41. Research Article
  42. Some inequalities of trigonometric approximation in weighted Orlicz spaces
  43. Research Article
  44. Summation methods applied to Voronovskaya-type theorems for the partial sums of Fourier series and for Fejér operators
  45. Research Article
  46. Dichotomies for Orlicz spaces
  47. Research Article
  48. On freely generated semigraph C*-algebras
  49. Research Article
  50. Operators with a given part of the numerical range
  51. Research Article
  52. Characterization of quasi-continuity of multifunctions of two variables
  53. Research Paper
  54. Confidence regions in singular weakly nonlinear regression models with constraints
  55. Research Article
  56. Global stability of an SEI model for plant diseases
https://doi.org/10.1515/ms-2015-0129
Keywords for this article
sixth order parabolic; time-periodic solution; existence

Articles in the same Issue

  1. Research Article
  2. Prof. RNDr. Ján Jakubík, DrSc. passed away
  3. Research Article
  4. On the number of slim, semimodular lattices
  5. Research Article
  6. Quantum computational algebra with a non-commutative generalization
  7. Research Article
  8. Lexico groups and direct products of lattice ordered groups
  9. Research Article
  10. Values of limit periodic sequences and functions
  11. Research Article
  12. On representation of quasi-Hopf group coalgebras
  13. Research Article
  14. On the q-Bernstein polynomials of the logarithmic function in the case q > 1
  15. Research Article
  16. On some modification of Darboux property
  17. Research Article
  18. Separating sets by peripherally continuous functions
  19. Research Article
  20. Persistence and extinction of a stochastic delay predator-prey model in a polluted environment
  21. Research Article
  22. On O’Malley preponderantly continuous functions
  23. Research Article
  24. Inequalities for the normalized Jensen functional with applications
  25. Research Article
  26. Majorization of starlike and convex functions of complex order involving linear operators
  27. Research Article
  28. Fekete–Szegö problem for some starlike functions related to shell-like curves
  29. Research Article
  30. Central functions for classes of concave univalent functions
  31. Research Article
  32. Inclusion properties for classes of analytic functions associated with conic domains
  33. Research Article
  34. Uniqueness of meromorphic functions and nonlinear differential polynomials sharing a nonzero polynomial
  35. Research Article
  36. Solutions and constrained null-controllability for a differential-difference equation
  37. Research Article
  38. Oscillation criteria for quasi-linear neutral delay dynamic equations on time scale
  39. Research Article
  40. Time periodic solutions for a sixth order nonlinear parabolic equation
  41. Research Article
  42. Some inequalities of trigonometric approximation in weighted Orlicz spaces
  43. Research Article
  44. Summation methods applied to Voronovskaya-type theorems for the partial sums of Fourier series and for Fejér operators
  45. Research Article
  46. Dichotomies for Orlicz spaces
  47. Research Article
  48. On freely generated semigraph C*-algebras
  49. Research Article
  50. Operators with a given part of the numerical range
  51. Research Article
  52. Characterization of quasi-continuity of multifunctions of two variables
  53. Research Paper
  54. Confidence regions in singular weakly nonlinear regression models with constraints
  55. Research Article
  56. Global stability of an SEI model for plant diseases
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 24.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0129/html
Scroll to top button