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Geometrical evolution of developed interfaces


Authors: Piero de Mottoni and Michelle Schatzman
Journal: Trans. Amer. Math. Soc. 347 (1995), 1533-1589
MSC: Primary 35B40; Secondary 35A30, 35K57, 58E12
DOI: https://doi.org/10.1090/S0002-9947-1995-1672406-7
MathSciNet review: 1672406
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Abstract: Consider the reaction-diffusion equation in $ {\mathbb{R}^N} \times {\mathbb{R}^ + }:{u_t} - {h^2}\Delta u + \varphi (u) = 0;\varphi $ is the derivative of a bistable potential with wells of equal depth and $ h$ is a small parameter. If the initial data has an interface, we give an asymptotic expansion of arbitrarily high order and error estimates valid up to time $ O({h^{ - 2}})$. At lowest order, the interface evolves normally, with a velocity proportional to the mean curvature.

Soit l'équation de réaction-diffusion dans $ {\mathbb{R}^N} \times {\mathbb{R}^ + },\quad {u_t} - {h^2}\Delta u + \varphi (u) = 0$, avec $ \varphi $ la dérivée d'un potentiel bistable à puits également profonds et $ h$ un petit paramètre. Pour une condition initiale possédant une interface, on donne un développement asymptotique d'ordre arbitrairement élevé, ainsi que des estimations d'erreur valides jusqu'à un temps en $ O({h^{ - 2}})$. A l'ordre le plus bas, l'interface évolue normalement, à une vitesse proportionnelle à la courbure moyenne.


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DOI: https://doi.org/10.1090/S0002-9947-1995-1672406-7
Article copyright: © Copyright 1995 American Mathematical Society

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