Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term

Yuan-Ming Wang

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 2, page 259-276
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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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 37.2 (2010): 259-276. <http://eudml.org/doc/194162>.

@article{Wang2010,
abstract = { 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. },
author = {Wang, Yuan-Ming},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Asymptotic behavior; finite difference equation; reaction diffusion equation; time delay; upper and lower solutions.; upper and lower solutions},
language = {eng},
month = {3},
number = {2},
pages = {259-276},
publisher = {EDP Sciences},
title = {Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term},
url = {http://eudml.org/doc/194162},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Wang, Yuan-Ming
TI - Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 2
SP - 259
EP - 276
AB - 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.
LA - eng
KW - Asymptotic behavior; finite difference equation; reaction diffusion equation; time delay; upper and lower solutions.; upper and lower solutions
UR - http://eudml.org/doc/194162
ER -

References

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  1. W.F. Ames, Numerical Methods for Partial Differential Equations. 3rd ed., Academic Press, San Diego (1992).  Zbl0759.65059
  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.  Zbl0325.35050
  3. A. Berman and R. Plemmons, Nonnegative Matrix in the Mathematical Science. Academic Press, New York (1979).  Zbl0484.15016
  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.  Zbl0383.35035
  5. G.E. Forsythe and W.R. Wasow, Finite Difference Methods for Partial Differential Equations. John Wiley, New York (1964).  Zbl0108.29803
  6. Y. Hamaya, On the asymptotic behavior of a diffusive epidemic model (AIDS). Nonlinear Anal.36 (1999) 685-696.  Zbl1005.92500
  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 Equations25 (1980) 113-127.  Zbl0427.35014
  8. X. Lu, Persistence and extinction in a competition-diffusion system with time delays. Canad. Appl. Math. Quart.2 (1994) 231-246.  Zbl0817.35043
  9. J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1976).  Zbl0949.65053
  10. C.V. Pao, Asymptotic behavior of solutions for finite-difference equations of reaction-diffusion. J. Math. Anal. Appl.144 (1989) 206-225.  Zbl0699.65070
  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.  Zbl0905.34063
  12. C.V. Pao, Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay. Numer. Methods Partial Differential Equations14 (1998) 339-351.  Zbl0919.65056
  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.  Zbl0933.65099
  14. C.V. Pao, Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992).  Zbl0777.35001
  15. C.V. Pao, Numerical methods for coupled systems of nonlinear parabolic boundary value problems. J. Math. Anal. Appl.151 (1990) 581-608.  Zbl0713.65053
  16. C.V. Pao, Numerical methods for systems of nonlinear parabolic equations with time delays. J. Math. Anal. Appl.240 (1999) 249-279.  Zbl0941.65083
  17. R.S. Varge, Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, NJ (1962).  
  18. Y. Yamada, Asymptotic behavior of solutions for semilinear Volterra diffusion equations. Nonlinear Anal.21 (1993) 227-239.  Zbl0806.35096
  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.  Zbl0914.92022

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