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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


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.
  • Liliane Alfonsi and Fred B. Weissler, Blow up in $\textbf {R}^n$ for a parabolic equation with a damping nonlinear gradient term, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 1–20. MR 1167826
  • Matania Ben-Artzi, Global existence and decay for a nonlinear parabolic equation, Nonlinear Anal. 19 (1992), no. 8, 763–768. MR 1186789, DOI 10.1016/0362-546X(92)90220-9
  • S. Benachour, B. Roynette, and P. Vallois, Asymptotic estimates of solutions of $u_t-{1\over 2}\Delta u=-|\nabla u|$ in $\textbf {R}_+\times \textbf {R}^d,\ d\geq 2$, J. Funct. Anal. 144 (1997), no. 2, 301–324. MR 1432587, DOI 10.1006/jfan.1996.2984
  • Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
  • David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, Vol. 224, Springer-Verlag, Berlin-New York, 1977. MR 0473443
  • Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
  • E. Shamir, private communication.
  • Philippe Souplet, Résultats d’explosion en temps fini pour une équation de la chaleur non linéaire, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 721–726 (French, with English and French summaries). MR 1354713
  • P. Souplet and F.B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Annales Inst. H. Poincaré – Anal. Nonlin. 16 (1999), 337–373.
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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:
  • 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:
  • MathSciNet review: 1615935