Remarks on a Nonlinear Parabolic Equation
HTML articles powered by AMS MathViewer
- by Matania Ben-Artzi, Jonathan Goodman and Arnon Levy
- Trans. Amer. Math. Soc. 352 (2000), 731-751
- DOI: https://doi.org/10.1090/S0002-9947-99-02336-3
- Published electronically: October 6, 1999
- PDF | 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
- 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.
Bibliographic 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