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

DOI:
https://doi.org/10.1090/S0002-9947-99-02336-3

Published electronically:
October 6, 1999

MathSciNet review:
1615935

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Abstract | References | Similar Articles | Additional Information

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

Article copyright:
© Copyright 1999
American Mathematical Society