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Volume 2,     Number 4,     Fall 1994

 

REACTION-DIFFUSION EQUATIONS
WITH INFINITE DELAY
SHIGUI RUAN AND JIANHONG WU

Abstract. We have developed several results on the existence and asymptotic behavior of mild solutions to reaction-diffusion systems that have infinite delays in the nonlinear reaction terms. We find that the semiflow generated by a cooperative and irreducible reaction-diffusion system with infinite delay is not compact but set-condensing, and not strongly order-preserving but quasi strongly order-preserving. These set-condenseness and quasi strong order-preserving properties allow us to use a modification, recently given by Freedman, Miller and one of the authors of this paper, of the well known monotone dynamical system theory due to Dancer, Hess, Hirsch, Matano, Smith, Thieme, Poláčik and Takáč to obtain some results about convergence and stability of solutions. Examples of Lotka-Volterra competition-diffusion models with distributed delay are given to illustrate the obtained results.

 

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