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Transactions of the American Mathematical Society

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



Length dependence of solutions of FitzHugh-Nagumo equations

Author: Clyde Collins
Journal: Trans. Amer. Math. Soc. 280 (1983), 809-832
MSC: Primary 35K57; Secondary 35B99, 92A09
MathSciNet review: 716852
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Abstract: 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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Article copyright: © Copyright 1983 American Mathematical Society