Existence result of global solutions for a class of generic reaction diffusion systems

Document Type : Research Paper

Authors

Fundamental and Numerical Mathematics Laboratory, Department of Mathematics, Faculty of Science, Ferhat Abbas University, Setif, Algeria

Abstract

In this paper, we prove the existence of weak global solutions for a class of generic reaction diffusion systems for which two main properties hold: the quasi-positivity and a triangular structure condition on the nonlinearities. The main result is a generalization of the work already done on models of a single reaction-diffusion equation. The model studied is applied in image recovery and contrast enhancement. It can also be applied to many models in biology and radiology.

Keywords

[1] A. Aarab, N. Alaa and H. Khalfi, Generic reaction diffusion model with application to image restoration and
enhancement, Elect. J. Diff. Eq. 125 (2018) 1–12.
[2] N. Alaa and M. Zirhem, Bio-inspired reaction diffusion system applied to image restoration, Int. J. Bio-Inspired
Comput. 12(2) (2018) 128–137.
[3] N. Alaa, M. Ait Oussous and Y. Ait Khouya, Anisotropic and nonlinear diffusion applied to image enhancement
and edge detection, Int. J. Computer Appl. Tech. 49(2) (2014) 122–133.
[4] N. Alaa, M. Ait Oussous, W. Bouarifi and D. Bensikaddour, Image restoration using a reaction diffusion process,
Elect. J. Diff. Eq. 197 (2014) 1–12.
[5] L. Alvarez, F. Guichard, P.L. Lions and J.M. Morel, Axioms and fundamental equations of image processing,
Arch. Rational Mech. Anal. 123 (1993) 199–257.
[6] L. Alvarez, P.L. Lions and J.M. Morel, Image selective smoothing and edge detection by nonlinear diffusion II,
SIAM J. Numerical Anal. 29(3) (1992) 845–866.
[7] H. Amann, Dynamic theory of quasilinear parabolic equations, ii. reaction diffusion systems, Differential Integral
Equations Vol. 3 (1990), no. 1, 13–75.
[8] P. Benilan and H. Brezis, Solutions faibles d’´equations d’´evolution dans les espaces de Hilbert, Ann. Inst. Fourier
(Grenoble), Tome 22(2) (1972) 311–329.
[9] F. Catt´e, P.L. Lions, J.M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion,
SIAM J. Numerical Anal. 29(1) (1992) 182–193.
[10] W.S. Du and Th. M. Rassias, Simultaneous generalizations of known fixed point theorems for a Meir-Keeler type
condition with applications, Int. J. Nonlinear Anal. Appl. 11(1) (2020) 55–66.
[11] J. Fr¨ohlich and J. Weicker, Image Processing Using a Wavelet Algorithm for Nonlinear Diffusion, Report 104,
Laboratory of Technomathematics, University of Kaiserslautern, Kaiserslautern, 1994.
[12] S.M.R. Hashemi, H. Hassanpour, E. Kozegar and T. Tan, Cystoscopic Image Classi cation Based on Combining
MLP and GA, Int. J. Nonlinear Anal. Appl. 11(1) (2020) 93–105.[13] S.M.R. Hashemi, H. Hassanpour, E. Kozegar and T. Tan, Cystoscopy image classification using deep convolutional
neural networks, Int. J. Nonlinear Anal. Appl. 10(1) (2019) 193–215.
[14] S. Kant and V. Kumar, Dynamical behavior of a stage structured prey-predator model, Int. J. Nonlinear Anal.
Appl. 7(1) (2016) 231–241.
[15] O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Lineinye i Kvazilineinye Uravneniya Parabolicheskogo
Tipa (Linear and Quasi-Linear Equations of Parabolic Type), Moscow, Nauka, 1967.
[16] J.L. Lions, Equations Diff´erentielles Operationnelles et Probl`emes aux Limites, vol. 111, Springer-Verlag, 2013.
[17] S. Mesbahi and N. Alaa, Mathematical analysis of a reaction diffusion model for image restoration, Ann. Univ.
Craiova, Math. Comp. Sci. Ser. 42(1) (2015) 70–79.
[18] S. Morfu, On some applications of diffusion processes for image processing, Phys. Lett. A 373(29) (2009) 24–44.
[19] J.D. Murray, Mathematical Biology I : An Introduction, Volume I, Springer-Verlag, 3rd edition, 2003.
[20] J.D. Murray, Mathematical Biology II : Spatial Models and Biochemical Applications, volume II, Springer-Verlag,
3rd edition, 2003.
[21] V. Pata, Fixed Point Theorems and Applications, Springer, 2019.
[22] M. Pierre, Global existence in reaction diffusion systems with dissipation of mass: a survey, Milan J. Math. 78(2)
(2010) 417–455.
[23] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, 2nd
ed., Springer, Birkh¨auser, 2019.
[24] D. Schmitt, Existence Globale ou Explosion Pour les Systemes De r´eaction-Diffusion Avec Contrˆole de Masse,
Ph.D. thesis, Nancy 1, 1995.
[25] J. Simon, Compact sets in the space L
p
(0, T, B), Ann. Math. Pura Appl. 146 (1986) 65–96.
[26] J. Weicker, Efficient image segmentation using partial differential equations and morphology, Pattern Recog. 34(9)
(2001) 1813-1824.
[27] J. Weicker and C. Schn¨orr, PDE-based preprocessing of medical images. Kunstliche Intelligenz, 3 (2000) 5–10.
[28] J. Weicker, Anisotropic Diffusion in Image Processing, PhD thesis, Universit¨at Kaiserslautern, Kaiserslautern,
Germany, 1996.
[29] S. Xu, S. Chenb and S. Aleksi´c, Fixed point theorems for generalized quasi-contractions in cone b-metric spaces
over Banach algebras without the assumption of normality with applications, Int. J. Nonlinear Anal. Appl. 8(2)
(2017) 335–353.
Volume 12, Special Issue
December 2021
Pages 663-676
  • Receive Date: 11 June 2020
  • Revise Date: 27 August 2021
  • Accept Date: 09 September 2021