Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, p. 259-276

This paper is concerned with the asymptotic behavior of the finite difference solutions of a class of nonlinear reaction diffusion equations with time delay. By introducing a pair of coupled upper and lower solutions, an existence result of the solution is given and an attractor of the solution is obtained without monotonicity assumptions on the nonlinear reaction function. This attractor is a sector between two coupled quasi-solutions of the corresponding “steady-state” problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that two coupled quasi-solutions coincide, is given. Also given is the application to a nonlinear reaction diffusion problem with time delay for three different types of reaction functions, including some numerical results which validate the theoretical analysis.

DOI : https://doi.org/10.1051/m2an:2003025
Classification:  35K57,  65M06,  74H40
Keywords: asymptotic behavior, finite difference equation, reaction diffusion equation, time delay, upper and lower solutions
@article{M2AN_2003__37_2_259_0,
     author = {Wang, Yuan-Ming},
     title = {Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {2},
     year = {2003},
     pages = {259-276},
     doi = {10.1051/m2an:2003025},
     zbl = {1026.35018},
     mrnumber = {1991200},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_2_259_0}
}
Wang, Yuan-Ming. Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 2, pp. 259-276. doi : 10.1051/m2an:2003025. http://www.numdam.org/item/M2AN_2003__37_2_259_0/

[1] W.F. Ames, Numerical Methods for Partial Differential Equations. 3rd ed., Academic Press, San Diego (1992). | MR 1184394 | Zbl 0759.65059

[2] D.G. Aronson and H.F. Weinberger, Nonlinear diffusion in population genetics, combustion and nerve propagation. Lecture Notes in Math. 446 (1975) 5-49. | Zbl 0325.35050

[3] A. Berman and R. Plemmons, Nonnegative Matrix in the Mathematical Science. Academic Press, New York (1979). | MR 544666 | Zbl 0484.15016

[4] E.D. Conway, D. Hoff and J.A. Smoller, Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Math. Appl. 35 (1978) 1-16. | Zbl 0383.35035

[5] G.E. Forsythe and W.R. Wasow, Finite Difference Methods for Partial Differential Equations. John Wiley, New York (1964). | MR 130124 | Zbl 0099.11103

[6] Y. Hamaya, On the asymptotic behavior of a diffusive epidemic model (AIDS). Nonlinear Anal. 36 (1999) 685-696. | Zbl 1005.92500

[7] A.W. Leung and D. Clark, Bifurcation and large time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data. J. Differential Equations 25 (1980) 113-127. | Zbl 0427.35014

[8] X. Lu, Persistence and extinction in a competition-diffusion system with time delays. Canad. Appl. Math. Quart. 2 (1994) 231-246. | Zbl 0817.35043

[9] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1976). | MR 273810 | Zbl 0241.65046

[10] C.V. Pao, Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion. J. Math. Anal. Appl. 144 (1989) 206-225. | Zbl 0699.65070

[11] C.V. Pao, Dynamics of a finite difference system of reaction diffusion equations with time delay. J. Differ. Equations Appl. 4 (1998) 1-11. | Zbl 0905.34063

[12] C.V. Pao, Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay. Numer. Methods Partial Differential Equations 14 (1998) 339-351. | Zbl 0919.65056

[13] C.V. Pao, Monotone methods for a finite difference system of reaction diffusion equation with time delay. Comput. Math. Appl. 36 (1998) 37-47. | Zbl 0933.65099

[14] C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992). | MR 1212084 | Zbl 0777.35001

[15] C.V. Pao, Numerical methods for coupled systems of nonlinear parabolic boundary value problems. J. Math. Anal. Appl. 151 (1990) 581-608. | Zbl 0713.65053

[16] C.V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays. J. Math. Anal. Appl. 240 (1999) 249-279. | Zbl 0941.65083

[17] R.S. Varge, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ (1962). | MR 158502 | Zbl 0133.08602

[18] Y. Yamada, Asymptotic behavior of solutions for semilinear Volterra diffusion equations. Nonlinear Anal. 21 (1993) 227-239. | Zbl 0806.35096

[19] Z.P. Yang and C.V. Pao, Positive solutions and dynamics of some reaction diffusion models in HIV transmission. Nonlinear Anal. 35 (1999) 323-341. | Zbl 0914.92022