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Article Dans Une Revue Mathematical Models and Methods in Applied Sciences Année : 2011

Asymptotic profiles for a travelling front solution of a biological equation

Romain Joly

Résumé

We are interested in the existence of depolarization waves in the human brain. These waves propagate in the grey matter and are absorbed in the white matter. We consider a two-dimensional model $u_t=\Delta u + f(u) \1_{|y|\leq R} - \alpha u \1_{|y|>R}$, with $f$ a bistable nonlinearity taking effect only on the domain $\Rm\times [-R,R]$, which represents the grey matter layer. We study the existence, the stability and the energy of non-trivial asymptotic profiles of possible travelling fronts. For this purpose, we present dynamical systems technics and graphic criteria based on Sturm-Liouville theory and apply them to the above equation. This yields three different behaviours of the solution $u$ after stimulation, depending of the thickness $R$ of the grey matter. This may partly explain the difficulties to observe depolarization waves in the human brain and the failure of several therapeutic trials.
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Dates et versions

hal-00482384 , version 1 (10-05-2010)

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Guillemette Chapuisat, Romain Joly. Asymptotic profiles for a travelling front solution of a biological equation. Mathematical Models and Methods in Applied Sciences, 2011, 21 (10), p. 2155-2177. ⟨10.1142/S0218202511005696⟩. ⟨hal-00482384⟩
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