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Asymptotic behavior of the numerical solutionsof time-delayed reaction diffusion equationswith non-monotone reaction term

Published online by Cambridge University Press:  15 November 2003

Yuan-Ming Wang
Yuan-Ming Wang*
Affiliation:
Department of Mathematics, East China Normal University, Shanghai 200062, China. [email protected].
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Abstract

This paper is concerned with the asymptotic behavior of thefinite 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 froma 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.

Keywords

Asymptotic behaviorfinite difference equationreaction diffusion equationtime delayupper and lower solutions.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 2 , March 2003 , pp. 259 - 276
Copyright
© EDP Sciences, SMAI, 2003

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References

W.F. Ames, Numerical Methods for Partial Differential Equations. 3rd ed., Academic Press, San Diego (1992).
Aronson, D.G. and Weinberger, H.F., Nonlinear diffusion in population genetics, combustion and nerve propagation. Lecture Notes in Math. 446 (1975) 5-49. CrossRef
A. Berman and R. Plemmons, Nonnegative Matrix in the Mathematical Science. Academic Press, New York (1979).
Conway, E.D., Hoff, D. and Smoller, J.A., Large time behavior of solutions of systems of nonlinear reaction-diffusion equations. SIAM J. Math. Appl. 35 (1978) 1-16. CrossRef
G.E. Forsythe and W.R. Wasow, Finite Difference Methods for Partial Differential Equations. John Wiley, New York (1964).
Hamaya, Y., On the asymptotic behavior of a diffusive epidemic model (AIDS). Nonlinear Anal. 36 (1999) 685-696. CrossRef
Leung, A.W. and Clark, D., Bifurcation and large time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data. J. Differential Equations 25 (1980) 113-127. CrossRef
Persistence, X. Lu and extinction in a competition-diffusion system with time delays. Canad. Appl. Math. Quart. 2 (1994) 231-246.
J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1976).
Pao, C.V., Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion. J. Math. Anal. Appl. 144 (1989) 206-225. CrossRef
Pao, C.V., Dynamics of a finite difference system of reaction diffusion equations with time delay. J. Differ. Equations Appl. 4 (1998) 1-11. CrossRef
Pao, C.V., Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay. Numer. Methods Partial Differential Equations 14 (1998) 339-351. 3.0.CO;2-N>CrossRef
Pao, C.V., Monotone methods for a finite difference system of reaction diffusion equation with time delay. Comput. Math. Appl. 36 (1998) 37-47. CrossRef
C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992).
Pao, C.V., Numerical methods for coupled systems of nonlinear parabolic boundary value problems. J. Math. Anal. Appl. 151 (1990) 581-608. CrossRef
Pao, C.V., Numerical methods for systems of nonlinear parabolic equations with time delays. J. Math. Anal. Appl. 240 (1999) 249-279. CrossRef
R.S. Varge, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ (1962).
Yamada, Y., Asymptotic behavior of solutions for semilinear Volterra diffusion equations. Nonlinear Anal. 21 (1993) 227-239. CrossRef
Yang, Z.P. and Pao, C.V., Positive solutions and dynamics of some reaction diffusion models in HIV transmission. Nonlinear Anal. 35 (1999) 323-341. CrossRef