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| We investigate an hyperbolic variation of the Allen--Cahn equation (bistable reaction--diffusion equation), where the Fick's law of diffusion is replaced by a relaxation term, thus introducing a delay in the process. The main feature of the model is the combination of the dissipation coming from the relaxation term, which play the role of the diffusive transport mechanism of the classical Allen--Cahn equation, and a zero--order reactive term, which determines the presence of two stable constant states.
Some rigorous results concerning existence and stability of traveling waves for this hyperbolic model are provided, together with numerical experiments, also in connection with the standard parabolic Allen--Cahn equation. |
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