May/June 2013 Convergence of a semidiscrete scheme for a forward-backward parabolic equation
Giovanni Bellettini, Carina Geldhauser, Matteo Novaga
Adv. Differential Equations 18(5/6): 495-522 (May/June 2013). DOI: 10.57262/ade/1363266255


We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation $u_t= (W'(u_x))_x$ with periodic boundary conditions in one space dimension, where $W$ is a standard double-well potential. We characterize the equation satisfied by the limit of the discretized solutions as the grid size goes to zero. Using an approximation argument, we show that it is possible to flow initial data ${\overline u}$ having regions where ${\overline u}_x$ falls within the concave region $\{W''<0\}$ of $W$, where the backward character of the equation manifests itself. It turns out that the limit equation depends on the way we approximate ${\overline u}$ in its unstable region. Our result can be viewed as a characterization, among all Young measure solutions of the equation, of the much smaller subset of those solutions which can be obtained as limit of the semidiscrete scheme.


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Giovanni Bellettini. Carina Geldhauser. Matteo Novaga. "Convergence of a semidiscrete scheme for a forward-backward parabolic equation." Adv. Differential Equations 18 (5/6) 495 - 522, May/June 2013.


Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1307.65118
MathSciNet: MR3086463
Digital Object Identifier: 10.57262/ade/1363266255

Primary: 35K20 , 35K55 , 65M12 , 65M60

Rights: Copyright © 2013 Khayyam Publishing, Inc.


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Vol.18 • No. 5/6 • May/June 2013
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