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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Geometrical evolution of developed interfaces
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by Piero de Mottoni and Michelle Schatzman PDF
Trans. Amer. Math. Soc. 347 (1995), 1533-1589 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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