Long-time behavior for a nonlinear fourth-order parabolic equation

Authors:
María J. Cáceres, J. A. Carrillo and G. Toscani

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1161-1175

MSC (2000):
Primary 35K65, 35B40, 35K35

Published electronically:
August 11, 2004

MathSciNet review:
2110436

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the asymptotic behavior of solutions of the initial- boundary value problem, with periodic boundary conditions, for a fourth-order nonlinear degenerate diffusion equation with a logarithmic nonlinearity. For strictly positive and suitably small initial data we show that a positive solution exponentially approaches its mean as time tends to infinity. These results are derived by analyzing the equation verified by the logarithm of the solution.

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Additional Information

**María J. Cáceres**

Affiliation:
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Email:
caceresg@ugr.es

**J. A. Carrillo**

Affiliation:
ICREA (Institució Catalana de Recerca i Estudis Avançats) and Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 - Bellaterra, Spain

Email:
carrillo@mat.uab.es

**G. Toscani**

Affiliation:
Department of Mathematics, University of Pavia, via Ferrata 1, 27100 Pavia, Italy

Email:
toscani@dimat.unipv.it

DOI:
https://doi.org/10.1090/S0002-9947-04-03528-7

Keywords:
Asymptotic behavior,
entropy dissipation,
degenerate parabolic equation,
diffusion equation

Received by editor(s):
November 13, 2002

Received by editor(s) in revised form:
October 2, 2003

Published electronically:
August 11, 2004

Additional Notes:
The authors were partially supported by the European IHP network “Hyperbolic and Kinetic Equations: Asymptotics, Numerics, Applications”, RNT2 2001 349 and Spanish-Italian bilateral HI01-175. The first and second authors acknowledge support from DGI-MCYT/FEDER project BFM2002-01710. The third author acknowledges partial support from the Italian Minister for Research, project “Mathematical Problems in Kinetic Theories”.

Article copyright:
© Copyright 2004
American Mathematical Society