August  2016, 21(6): 1839-1858. doi: 10.3934/dcdsb.2016025

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

1. 

College of Mathematics and Computational Science, Shenzhen University, 518060 Shenzhen, China

2. 

Department of Mathematics, Harbin Institute of Technology, 150001 Harbin, China

3. 

School of Mathematics, Jilin University, Changchun 130012

Received  December 2014 Revised  May 2016 Published  June 2016

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.
Citation: Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025
References:
[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.

show all references

References:
[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.

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