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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Length dependence of solutions of FitzHugh-Nagumo equations
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by Clyde Collins PDF
Trans. Amer. Math. Soc. 280 (1983), 809-832 Request permission


We investigate the behavior of the solutions of the problem \[ \begin {array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsilon + {u^2}(1 + \alpha - u),} & {{\upsilon _t} = \eta {\upsilon _{xx}} + \sigma u - \gamma \upsilon ,} \\ {u(0,t) = g(t),\quad \upsilon (0,t) = h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \\ \end {array} \] where $t \geqslant 0$ and $0 < x < L \leqslant \infty$. Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for $L$ infinite and the solutions when $L$ is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 809-832
  • MSC: Primary 35K57; Secondary 35B99, 92A09
  • DOI:
  • MathSciNet review: 716852