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Pré-Publication, Document De Travail Année : 2007

The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system

Résumé

We consider an Allen-Cahn type equation with a bistable nonlinearity associated to a double-well potential whose well-depths can be slightly unbalanced, and where the coefficient of the nonlinear reaction term is very small. Given rather general initial data, we perform a rigorous analysis of both the generation and the motion of interface. More precisely we show that the solution develops a steep transition layer within a small time, and we present an optimal estimate for its width. We then consider a class of reaction-diffusion systems which includes the FitzHugh-Nagumo system as a special case. Given rather general initial data, we show that the first component of the solution vector develops a steep transition layer and that all the results mentioned above remain true for this component.
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Dates et versions

hal-00174999 , version 1 (26-09-2007)
hal-00174999 , version 2 (17-09-2009)

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Matthieu Alfaro, Danielle Hilhorst, Hiroshi Matano. The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system. 2007. ⟨hal-00174999v1⟩
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