# American Institute of Mathematical Sciences

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 & 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. doi: 10.1016/j.camwa.2008.01.017. Google Scholar [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. doi: 10.1016/j.jde.2008.02.022. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9947-99-01981-9. Google Scholar [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. doi: 10.4171/IFB/151. Google Scholar [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. doi: 10.1016/j.jde.2010.05.011. Google Scholar [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. Google Scholar [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. doi: 10.1017/S0308210500026986. Google Scholar [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. doi: 10.1006/jdeq.2000.4013. Google Scholar [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. doi: 10.1007/BF01766148. Google Scholar [11] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383. doi: 10.1137/050624522. Google Scholar [12] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2011). doi: 10.1007/978-3-642-18363-8. Google Scholar [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. doi: 10.2307/1971423. Google Scholar [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. doi: 10.1007/s00030-007-5018-z. Google Scholar [15] R. C. Gonzalez and R. E. Woods, Digital Image Processing,, $2^{nd}$ edition, (2002). Google Scholar [16] Y. Gousseau and J. M. Morel, Are natural images of bounded variation?,, SIAM Journal on Mathematical Analysis, 33 (2001), 634. doi: 10.1137/S0036141000371150. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.04.015. Google Scholar [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. doi: 10.1109/TIP.2011.2169272. Google Scholar [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. doi: 10.1016/j.mcm.2010.12.031. Google Scholar [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. doi: 10.1017/S0308210500020242. Google Scholar [21] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society: Providence, (1968). Google Scholar [22] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar [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. doi: 10.1007/s13370-011-0030-1. Google Scholar [24] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations,, Ann. Mat. Pura ed Applicata, 177 (1999), 143. doi: 10.1007/BF02505907. Google Scholar [25] A. Porretta, Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum,, Dynam. Systems Appl., 7 (1998), 53. Google Scholar [26] H. Redwane, Existence of a solution for a class of nonlinear parabolic systems,, Electron. J. Qual. Theory Differ. Equ, 24 (2007), 1. Google Scholar [27] S. Segura de Lenón and J. Toledo, Regularity for entropy solutions of parabolic $p$-Laplacian type equations,, Publ. Mat., 43 (1999), 665. doi: 10.5565/PUBLMAT_43299_08. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2009.11.024. Google Scholar

show all references

##### References:
 [1] R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing,, Comput. Math. Appl., 56 (2008), 874. doi: 10.1016/j.camwa.2008.01.017. Google Scholar [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. doi: 10.1016/j.jde.2008.02.022. Google Scholar [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. Google Scholar [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. doi: 10.1090/S0002-9947-99-01981-9. Google Scholar [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. doi: 10.4171/IFB/151. Google Scholar [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. doi: 10.1016/j.jde.2010.05.011. Google Scholar [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. Google Scholar [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. doi: 10.1017/S0308210500026986. Google Scholar [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. doi: 10.1006/jdeq.2000.4013. Google Scholar [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. doi: 10.1007/BF01766148. Google Scholar [11] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383. doi: 10.1137/050624522. Google Scholar [12] L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents,, Lecture Notes in Mathematics, (2011). doi: 10.1007/978-3-642-18363-8. Google Scholar [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. doi: 10.2307/1971423. Google Scholar [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. doi: 10.1007/s00030-007-5018-z. Google Scholar [15] R. C. Gonzalez and R. E. Woods, Digital Image Processing,, $2^{nd}$ edition, (2002). Google Scholar [16] Y. Gousseau and J. M. Morel, Are natural images of bounded variation?,, SIAM Journal on Mathematical Analysis, 33 (2001), 634. doi: 10.1137/S0036141000371150. Google Scholar [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. doi: 10.1016/j.nonrwa.2011.04.015. Google Scholar [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. doi: 10.1109/TIP.2011.2169272. Google Scholar [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. doi: 10.1016/j.mcm.2010.12.031. Google Scholar [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. doi: 10.1017/S0308210500020242. Google Scholar [21] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type,, American Mathematical Society: Providence, (1968). Google Scholar [22] J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,, Dunod, (1969). Google Scholar [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. doi: 10.1007/s13370-011-0030-1. Google Scholar [24] A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations,, Ann. Mat. Pura ed Applicata, 177 (1999), 143. doi: 10.1007/BF02505907. Google Scholar [25] A. Porretta, Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum,, Dynam. Systems Appl., 7 (1998), 53. Google Scholar [26] H. Redwane, Existence of a solution for a class of nonlinear parabolic systems,, Electron. J. Qual. Theory Differ. Equ, 24 (2007), 1. Google Scholar [27] S. Segura de Lenón and J. Toledo, Regularity for entropy solutions of parabolic $p$-Laplacian type equations,, Publ. Mat., 43 (1999), 665. doi: 10.5565/PUBLMAT_43299_08. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.jde.2009.11.024. Google Scholar
 [1] Mostafa Bendahmane, Kenneth H. Karlsen. Renormalized solutions of an anisotropic reaction-diffusion-advection system with $L^1$ data. Communications on Pure & Applied Analysis, 2006, 5 (4) : 733-762. doi: 10.3934/cpaa.2006.5.733 [2] Takashi Kajiwara. A Heteroclinic Solution to a Variational Problem Corresponding to FitzHugh-Nagumo type Reaction-Diffusion System with Heterogeneity. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2133-2156. doi: 10.3934/cpaa.2017106 [3] Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631 [4] Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191 [5] Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 [6] Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493 [7] Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169 [8] Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227 [9] W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893 [10] José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299 [11] Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039 [12] Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673 [13] Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817 [14] José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85 [15] Bo Li, Xiaoyan Zhang. Steady states of a Sel'kov-Schnakenberg reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1009-1023. doi: 10.3934/dcdss.2017053 [16] Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182 [17] Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks & Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369 [18] Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032 [19] Hiroshi Matsuzawa. On a solution with transition layers for a bistable reaction-diffusion equation with spatially heterogeneous environments. Conference Publications, 2009, 2009 (Special) : 516-525. doi: 10.3934/proc.2009.2009.516 [20] Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041

2018 Impact Factor: 1.008

## Metrics

• PDF downloads (15)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]