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Convergence of a semidiscrete scheme for a forward-backward parabolic equation

Bellettini, Giovanni ; Geldhauser, Carina LU orcid and Novaga, Matteo (2013) In Advances in Differential Equations 18(5-6). p.495-522
Abstract

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation ut = (W?(ux))x with periodic boundary conditions in one space dimension, where W is a standard double-well potential. We characterize the equation satised 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 ū having regions where ūx 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 ū in its unstable region. Our result can be viewed as a characterization, among all... (More)

We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation ut = (W?(ux))x with periodic boundary conditions in one space dimension, where W is a standard double-well potential. We characterize the equation satised 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 ū having regions where ūx 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 ū 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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type
Contribution to journal
publication status
published
subject
in
Advances in Differential Equations
volume
18
issue
5-6
pages
28 pages
publisher
Khayyam Publishing, Inc.
external identifiers
  • scopus:84884170583
ISSN
1079-9389
language
English
LU publication?
no
id
fb5cae6f-4fb1-456d-8b89-07c49f15a80d
date added to LUP
2021-02-08 12:11:51
date last changed
2025-04-04 14:44:13
@article{fb5cae6f-4fb1-456d-8b89-07c49f15a80d,
  abstract     = {{<p>We study the convergence of a semidiscrete scheme for the forward-backward parabolic equation u<sub>t</sub> = (W?(u<sub>x</sub>))<sub>x</sub> with periodic boundary conditions in one space dimension, where W is a standard double-well potential. We characterize the equation satised 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 ū having regions where ū<sub>x</sub> falls within the concave region -W" &lt; 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 ū 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.</p>}},
  author       = {{Bellettini, Giovanni and Geldhauser, Carina and Novaga, Matteo}},
  issn         = {{1079-9389}},
  language     = {{eng}},
  number       = {{5-6}},
  pages        = {{495--522}},
  publisher    = {{Khayyam Publishing, Inc.}},
  series       = {{Advances in Differential Equations}},
  title        = {{Convergence of a semidiscrete scheme for a forward-backward parabolic equation}},
  volume       = {{18}},
  year         = {{2013}},
}