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Transactions of the American Mathematical Society

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Remarks on a Nonlinear Parabolic Equation


Authors: Matania Ben-Artzi, Jonathan Goodman and Arnon Levy
Journal: Trans. Amer. Math. Soc. 352 (2000), 731-751
MSC (1991): Primary 35K15, 35K55
Published electronically: October 6, 1999
MathSciNet review: 1615935
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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
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

DOI: https://doi.org/10.1090/S0002-9947-99-02336-3
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
Article copyright: © Copyright 1999 American Mathematical Society