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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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Remarks on a Nonlinear Parabolic Equation
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by Matania Ben-Artzi, Jonathan Goodman and Arnon Levy PDF
Trans. Amer. Math. Soc. 352 (2000), 731-751 Request permission

Abstract:

The equation $u_{t} =\Delta u +\mu |\nabla u |$, $\mu \in \mathbb {R}$, is studied in $\mathbb {R}^{n}$ and in the periodic case. It is shown that the equation is well-posed in $L^{1}$ and possesses regularizing properties. For nonnegative initial data and $\mu <0$ the solution decays in $L^{1}(\mathbb {R}^{n})$ as $t\to \infty$. In the periodic case it tends uniformly to a limit. A consistent difference scheme is presented and proved to be stable and convergent.
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Additional Information
  • Matania Ben-Artzi
  • Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
  • MR Author ID: 34290
  • ORCID: 0000-0002-6782-4085
  • Email: mbartzi@math.huji.ac.il
  • Jonathan Goodman
  • Affiliation: Courant Institute of Mathematical Sciences, New York, New York 10012
  • Email: goodman@cims.nyu.ed
  • Arnon Levy
  • Affiliation: Courant Institute of Mathematical Sciences, New York, New York 10012
  • Received by editor(s): November 11, 1996
  • Received by editor(s) in revised form: September 22, 1997
  • Published electronically: October 6, 1999
  • Additional Notes: The first author was partially supported by a grant from the Israel Science Foundation
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 731-751
  • MSC (1991): Primary 35K15, 35K55
  • DOI: https://doi.org/10.1090/S0002-9947-99-02336-3
  • MathSciNet review: 1615935