## 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.## References

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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