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.

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/

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