Some Properties of Solutions of a Fourth-Order Parabolic Equation for Image Processing

  • Changchun LiuEmail author
  • Manli Jin


In this paper, for the IBVP of a fourth-order nonlinear parabolic equation, which is related to image analysis, we studied the existence and uniqueness of weak solutions. Moreover, we also considered the asymptotic behavior and the regularity of solutions of such problem.


Fourth-order parabolic equation Existence Asymptotic behavior Regularity 

Mathematics Subject Classification

35D05 35B40 35G30 35K55 


  1. 1.
    Bernis, F.: Qualitative properties for some nonlinear higher order degenerate parabolic equations. Houst. J. Math. 14(3), 319–352 (1988)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chang, K.: Critical Point Theory and its Applications. Shanghai Science and Technology Press, Shanghai (1986)zbMATHGoogle Scholar
  3. 3.
    Greer, J.B., Bertozzi, A.L.: \(H^1\) solutions of a class of fourth order nonlinear equations for image processing. Discrete Contin. Dyn. Syst. 10(1–2), 349–366 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Hao, A., Zhou, J.: Blowup, extinction and non-extinction for a nonlocal p-biharmonic parabolic equation. Appl. Math. Lett. 64, 198–204 (2017)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaire, Dunod et Gauthier-Villars (1969)Google Scholar
  6. 6.
    Liu, C., Yin, J., Gao, H.: A generalized thin film equation. Chin. Ann. Math. 25 B(3), 347–358 (2004)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Osher, S., Solé, A., Vese, L.: Image decomposition and restoration using total variation minimization and the \(H^{-1}\)norm. Multiscale Model. Simul. 1(3), 349–370 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  9. 9.
    Tumblin, J., Turk, G.: A boundary hierarchy for detail-preserving contrast reduction, In: Processings of the SIGGRAPH 1999 Annual Conference on Computer Graphics, Los Angeles, CA, USA, pp. 83–90 (1999)Google Scholar
  10. 10.
    Wang, L., Zhang, C., Zhou, S.: Existence and uniqueness of weak solutions for a 2D low-curvature equation. J. Math. Anal. Appl. 452, 297–311 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Wei, G.W.: Genenralized Perona–Malik equation for image processing. IEEE Signal Process. Lett. 6(7), 165–167 (1999)CrossRefGoogle Scholar
  12. 12.
    Xu, M., Zhou, S.: Existence and uniqueness of weak solutions for a fourth-order nonlinear parabolic equation. J. Math. Anal. Appl. 325, 636–654 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    You, Y.L., Kaveh, M.: Fourth-order partial differential equations for noise removal. IEEE Trans. Image Process. 9(10), 1723–1730 (2000)MathSciNetCrossRefGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of MathematicsJilin UniversityChangchunChina

Personalised recommendations