- Research Article
- Open access
- Published:
Self-similar singular solution of doubly singular parabolic equation with gradient absorption term
Journal of Inequalities and Applications volume 2006, Article number: 20438 (2006)
Abstract
We deal with the self-similar singular solution of doubly singular parabolic equation with a gradient absorption term for, and in. By shooting and phase plane methods, we prove that when there exists self-similar singular solution, while there is no any self-similar singular solution. In case of existence, the self-similar singular solution is the self-similar very singular solutions which have compact support. Moreover, the interface relation is obtained.
References
Brézis H, Friedman A: Nonlinear parabolic equations involving measures as initial conditions. Journal de Mathématiques Pures et Appliquées. Neuvième Série 1983,62(1):73–97.
Brézis H, Peletier LA, Terman D: A very singular solution of the heat equation with absorption. Archive for Rational Mechanics and Analysis 1986,95(3):185–209.
Chen X, Qi Y, Wang M: Long time behavior of solutions to-Laplacian equation with absorption. SIAM Journal on Mathematical Analysis 2003,35(1):123–134.
Chen X, Qi Y, Wang M: Self-similar singular solutions of a-Laplacian evolution equation with absorption. Journal of Differential Equations 2003,190(1):1–15. 10.1016/S0022-0396(02)00039-6
Escobedo M, Kavian O, Matano H: Large time behavior of solutions of a dissipative semilinear heat equation. Communications in Partial Differential Equations 1995,20(7–8):1427–1452. 10.1080/03605309508821138
Herraiz L: Asymptotic behaviour of solutions of some semilinear parabolic problems. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1999,16(1):49–105.
Kamin S, Vázquez JL: Singular solutions of some nonlinear parabolic equations. Journal d'Analyse Mathématique 1992, 59: 51–74.
Kwak M: A porous media equation with absorption. I. Long time behaviour. Journal of Mathematical Analysis and Applications 1998,223(1):96–110. 10.1006/jmaa.1998.5961
Leoni G: A very singular solution for the porous media equationwhen. Journal of Differential Equations 1996,132(2):353–376. 10.1006/jdeq.1996.0184
Leoni G: On very singular self-similar solutions for the porous media equation with absorption. Differential and Integral Equations. An International Journal for Theory & Applications 1997,10(6):1123–1140.
Peletier LA, Wang JY: A very singular solution of a quasilinear degenerate diffusion equation with absorption. Transactions of the American Mathematical Society 1988,307(2):813–826. 10.1090/S0002-9947-1988-0940229-6
Peletier LA, Zhao JN: Large time behaviour of solutions of the porous media equation with absorption: the fast diffusion case. Nonlinear Analysis 1991,17(10):991–1009. 10.1016/0362-546X(91)90059-A
Qi Y, Wang M: Singular solutions of doubly singular parabolic equations with absorption. Electronic Journal of Differential Equations 2000,2000(67):22.
Qi Y, Wang M: The self-similar profiles of generalized KPZ equation. Pacific Journal of Mathematics 2001,201(1):223–240. 10.2140/pjm.2001.201.223
Shi P: Self-similar singular solution of a-Laplacian evolution equation with gradient absorption term. Journal of Partial Differential Equations 2004,17(4):369–383.
Zhao JN: The asymptotic behaviour of solutions of a quasilinear degenerate parabolic equation. Journal of Differential Equations 1993,102(1):33–52. 10.1006/jdeq.1993.1020
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Shi, P., Wang, M. Self-similar singular solution of doubly singular parabolic equation with gradient absorption term. J Inequal Appl 2006, 20438 (2006). https://doi.org/10.1155/JIA/2006/20438
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/JIA/2006/20438