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Wave solutions for a discrete reaction-diffusion equation

Published online by Cambridge University Press:  23 October 2000

A. CARPIO
Affiliation:
Department of Applied Mathematics, Universidad Complutense de Madrid, Madrid, Spain
S. J. CHAPMAN
Affiliation:
Mathematical Institute , 24–29 St. Giles, Oxford OX1 3LB, UK
S. HASTINGS
Affiliation:
Department of Mathematics, University of Pittsburgh, PA, USA
J. B. McLEOD
Affiliation:
Department of Mathematics, University of Pittsburgh, PA, USA

Abstract

Motivated by models from fracture mechanics and from biology, we study the infinite system of differential equations

formula here

where A and F are positive parameters. For fixed A > 0 we show that there are monotone travelling waves for F in an interval Fcrit < F < A, and we are able to give a rigorous upper bound for Fcrit, in contrast to previous work on similar problems. We raise the problem of characterizing those nonlinearities (apparently the more common) for which Fcrit > 0. We show that, for the sine nonlinearity, this is true if A > 2. (Our method yields better estimates than this, but does not include all A > 0.) We also consider the existence and multiplicity of time independent solutions when |F|< Fcrit.

Type
Research Article
Copyright
2000 Cambridge University Press

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