Asymptotic Behavior of Some Parabolic Equations and Application in Image Restoration

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Fatima Zohra Zeghbib, Abir Bounaama, Messaoud Maouni, Halim Zeghdoudi


In this paper, we consider some nonlinear parabolic problem involving the well known p-laplacian and some operator having exponential growth with respect to the gradient. We start by dealing the asymptotic behavior for some evolution equation then we give some numerical results with an application in image processing.

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  1. R. Aboulaich, D. Meskine, A. Souissi, New Diffusion Models in Image Processing, Computers Math. Appl. 56 (2008), 874–882.
  2. R. Adams, Sobolev Spaces, Academic Press, New York, 1975.
  3. H. Alaa, N.E. Alaa, A. Bouchriti, A. Charkaoui, An Improved Nonlinear Anisotropic PDE with p(x)-Growth Conditions Applied to Image Restoration and Enhancement, Preprint. (2022).
  4. L. Alvarez, L. Mazorra, Signal and Image Restoration Using Shock Filters and Anisotropic Diffusion, SIAM J. Numer. Anal. 31 (1994), 590–605.
  5. A. Atlas, F. Karami, D. Meskine, The Perona–malik Inequality and Application to Image Denoising, Nonlinear Anal.: Real World Appl. 18 (2014), 57–68.
  6. G. Aubert, P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Springer, New York, 2006.
  7. A. Chambolle, P.L. Lions, Image Recovery via Total Variation Minimization and Related Problems, Numer. Math. 76 (1997), 167-188.
  8. Y. Chen, S. Levine, M. Rao, Variable Exponent, Linear Growth Functionals in Image Restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.
  9. U. Diewald, T. Preusser, M. Rumpf, R. Strzodka, Diffusion Models and Their Accelerated Solution in Image and Surface Processing, Acta Math. Univ. Comenianae, 70 (2001), 15-34.
  10. A. Elmahi, D. Meskine, Strongly Nonlinear Parabolic Equations With Natural Growth Terms in Orlicz Spaces, Nonlinear Anal.: Theory Methods Appl. 60 (2005), 1–35.
  11. A. Elmahi, D. Meskine, Parabolic Equations in Orlicz Spaces, J. London Math. Soc. 72 (2005), 410–428.
  12. J.P. Gossez, Some Approximation Properties in Orlicz-Sobolev Spaces, Stud. Math. 74 (1982), 17-24.
  13. P. Harjulehto, P. Hästö, V. Latvala, O. Toivanen, Critical Variable Exponent Functionals in Image Restoration, Appl. Math. Lett. 26 (2013), 56–60.
  14. M. Kbiri Alaoui, D. Meskine, A. Souissi, On Some Class of Nonlinear Parabolic Inequalities in Orlicz Spaces, Nonlinear Anal.: Theory Methods Appl. 74 (2011), 5863–5875.
  15. P. Kogut, Y. Kohut, R. Manzo, Existence Result and Approximation of an Optimal Control Problem for the Perona–malik Equation, Ricerche Mat. (2022).
  16. R. Landes, On the Existence of Weak Solutions for Quasilinear Parabolic Initial-Boundary Value Problems, Proc. R. Soc. Edinburgh: Sect. A Math. 89 (1981), 217-237.
  17. S. Lecheheb, M. Maouni, H. Lakhal, Existence of the Solution of a Quasilinear Equation and Its Application to Image Denoising, Int. J. Computer Sci. Commun. Inform. Technol. 7 (2019), 1-6.
  18. S. Lecheheb, M. Maouni, H. Lakhal, Image Restoration Using Nonlinear Eliptic Equation, Int. J. Computer Sci. Computer Sci. Commun. Inform. Technol. 6 (2019), 32-37.
  19. H. Matallah, M. Maouni, H. Lakhal, Image Restoration by a Fractional Reaction-Diffusion Process, Int. J. Anal. Appl. 19 (2021), 709-724.
  20. M. Maouni, F.Z. Nouri, Image Restoration Based on p-Gradient Model, Int. J. Appl. Math. Stat. 41 (2013), 48-57.
  21. P. Perona, J. Malik, Scale-Space and Edge Detection Using Anisotropic Diffusion, IEEE Trans. Pattern Anal. Mach. Intell. 12 (1990), 629–639.
  22. L.I. Rudin, S. Osher, E. Fatemi, Nonlinear Total Variation Based Noise Removal Algorithms, Physica D: Nonlinear Phenomena. 60 (1992), 259-268.
  23. W. Walter, Differential and Integral Inequalities, Springer, Berlin, New York, (1970).
  24. Z.F. Zohra, M. Messaoud, N.F. Zohra, Overlapping and Nonoverlapping Domain Decomposition Methods for Image Restoration, Int. J. Appl. Math. Stat. 40 (2013), 123-128.
  25. Z.F. Zohra, M. Maouni, Image Processing by a Fractional Partial Differential Equation, Int. J. Computer Sci. Commun. Inform. Technol. 7 (2019), 13-16.