Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons
Authors:
Jonathan Bell and Chris Cosner
Journal:
Quart. Appl. Math. 42 (1984), 1-14
MSC:
Primary 92A10; Secondary 34K15
DOI:
https://doi.org/10.1090/qam/736501
MathSciNet review:
736501
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Abstract: Using a comparison theorem technique, we study the long time behavior of certain classes of nonlinear difference-differential systems. Zero is a solution for these systems. We are concerned in this paper with conditions forcing nonconvergence to zero of solutions as time approaches infinity; that is, we obtain threshold properties of the systems. The results parallel results by Aronson and Weinberger on reaction-diffusion equations somewhat, and the study was motivated by consideration of models for myelinated nerve axons.
- D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Springer, Berlin, 1975, pp. 5–49. Lecture Notes in Math., Vol. 446. MR 0427837
- Jonathan Bell, Some threshold results for models of myelinated nerves, Math. Biosci. 54 (1981), no. 3-4, 181–190. MR 630848, DOI https://doi.org/10.1016/0025-5564%2881%2990085-7
H. Cohen, Nonlinear diffusion models, Studies in Applied Mathematics, ed. A. Taub, Math. Assoc. Amer., 1971
P. Morell and W. T. Norton, Myelin, Sci. Amer. 242 (1980), 88–118
J. Rinzel, Integration and propagation of neuroelectric signals, Studies in Mathematical Biology, ed. S. A. Levin, Math. Assoc. Amer., 1977
A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys. 26 (1964), 247–254
- Wolfgang Walter, Differential and integral inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. MR 0271508
D. G. Aronson and H. F. Weinberger: Nonlinear diffusion in population genetics, combustion, and nerve propagation, Proceedings of the Tulane Program in Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446, Springer-Verlag, New York, 1975
J. Bell, Some threshold results for models of myelinated nerves, Math. Biosciences 54 (1981), 181–190
H. Cohen, Nonlinear diffusion models, Studies in Applied Mathematics, ed. A. Taub, Math. Assoc. Amer., 1971
P. Morell and W. T. Norton, Myelin, Sci. Amer. 242 (1980), 88–118
J. Rinzel, Integration and propagation of neuroelectric signals, Studies in Mathematical Biology, ed. S. A. Levin, Math. Assoc. Amer., 1977
A. C. Scott, Analysis of a myelinated nerve model, Bull. Math. Biophys. 26 (1964), 247–254
W. Walter, Differential and integral inequalities, Springer-Verlag, New York, 1970
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Article copyright:
© Copyright 1984
American Mathematical Society
