\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Renormalized solutions to a reaction-diffusion system applied to image denoising

Abstract Related Papers Cited by
  • This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.
    Mathematics Subject Classification: 35K65, 35D30.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.doi: 10.1016/j.camwa.2008.01.017.

    [2]

    F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary condtions and $L^1$-data, Journal of differential equations, 244 (2008), 2764-2803.doi: 10.1016/j.jde.2008.02.022.

    [3]

    F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Advances in mathematical sciences and applications, 7 (1997), 183-213.

    [4]

    F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with $L^1$ data, Trans. Amer. Math. Soc, 351 (1999), 285-306.doi: 10.1090/S0002-9947-99-01981-9.

    [5]

    F. Andreu, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces and Free Boundaries, 8 (2006), 447-479.doi: 10.4171/IFB/151.

    [6]

    M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$-data, J. Differential Equations, 249 (2010), 1483-1515.doi: 10.1016/j.jde.2010.05.011.

    [7]

    P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273.

    [8]

    D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with $L^1$ data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.doi: 10.1017/S0308210500026986.

    [9]

    D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.doi: 10.1006/jdeq.2000.4013.

    [10]

    L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1998), 183-196.doi: 10.1007/BF01766148.

    [11]

    Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.doi: 10.1137/050624522.

    [12]

    L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011.doi: 10.1007/978-3-642-18363-8.

    [13]

    R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.doi: 10.2307/1971423.

    [14]

    J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, Nonlinear Differential Equations Appl., 14 (2007), 181-205.doi: 10.1007/s00030-007-5018-z.

    [15]

    R. C. Gonzalez and R. E. Woods, Digital Image Processing, $2^{nd}$ edition, Pearson Prentice Hall, 2002.

    [16]

    Y. Gousseau and J. M. Morel, Are natural images of bounded variation?, SIAM Journal on Mathematical Analysis, 33 (2001), 634-648.doi: 10.1137/S0036141000371150.

    [17]

    Z. C. Guo, Q. Liu, J. B. Sun and B. Y. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Analysis: Real World Applications, 12 (2011), 2904-2918.doi: 10.1016/j.nonrwa.2011.04.015.

    [18]

    Z. C. Guo, J. B. Sun, D. Z. Zhang and B. Y. Wu, Adaptive Perona-Malik model based on the variable exponent for image denoising, IEEE Transactions on Image Processing, 21 (2012), 958-967.doi: 10.1109/TIP.2011.2169272.

    [19]

    Z. C. Guo, J. X. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Mathematical and Computer Modelling, 53 (2011), 1336-1350.doi: 10.1016/j.mcm.2010.12.031.

    [20]

    R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect A, 89 (1981), 217-237.doi: 10.1017/S0308210500020242.

    [21]

    O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1968.

    [22]

    J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.

    [23]

    I. Nyanquini and S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afrika Matematika, 23 (2012), 205-228.doi: 10.1007/s13370-011-0030-1.

    [24]

    A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata, 177 (1999), 143-172.doi: 10.1007/BF02505907.

    [25]

    A. Porretta, Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum, Dynam. Systems Appl., 7 (1998), 53-71.

    [26]

    H. Redwane, Existence of a solution for a class of nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ, 24 (2007), 1-18.

    [27]

    S. Segura de Lenón and J. Toledo, Regularity for entropy solutions of parabolic $p$-Laplacian type equations, Publ. Mat., 43 (1999), 665-683.doi: 10.5565/PUBLMAT_43299_08.

    [28]

    Z. Q. Wu, J. Y. Yin, H. L. Li and J. N. Zhao, Nonlinear Diffusion Equations, World Scientific Publishing Company, 2001.doi: 10.1142/9789812799791.

    [29]

    C. Zhang and S. L. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$-data, J. Differential Equations, 248 (2010), 1376-1400.doi: 10.1016/j.jde.2009.11.024.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(282) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return