Asymptotic effects of boundary perturbations in  excitable systems

Monica De Angelis; Pasquale Renno

doi:10.3934/dcdsb.2014.19.2039

Article Contents

2014, Volume 19, Issue 7: 2039-2045. Doi: 10.3934/dcdsb.2014.19.2039

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Asymptotic effects of boundary perturbations in excitable systems

Monica De Angelis^1, and
Pasquale Renno^1,

1.
University of Naples Federico II, Via Claudio, 21, Naples, 80121

Received: March 31, 2013

Revised: February 28, 2014

Published: August 2014

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Abstract

A Neumann problem in the strip for the Fitzhugh Nagumo system is considered. The transformation in a non linear integral equation permits to deduce a priori estimates for the solution. A complete asymptotic analysis shows that for large $ t $ the effects of the initial data vanish while the effects of boundary disturbances $ \varphi_1 (t), $ $ \varphi_2(t) $ depend on the properties of the data. When $ \varphi_1,\,\, \varphi_2 $ are convergent for large $ t $, the solution is everywhere bounded and depends on the asymptotic values of $ \varphi_1 , $ $ \varphi_2 $. More, when $ \varphi_i \in L^1 (0,\infty) (i=1,2)$ too, the effects are vanishing.

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Mathematics Subject Classification: 44A10, 35K57, 35A08.

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