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

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

Published electronically:
August 11, 2004

MathSciNet review:
2110436

Full-text PDF

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.

**1.**P. M. Bleher, J. L. Lebowitz, E. R. Speer,*Existence and positivity of solutions of a fourth order nonlinear PDE describing interface fluctuations*, Comm. Pure Appl. Math.**47**, 923-942 (1994). MR**95e:35173****2.**J. A. Carrillo, A. Jüngel, P. Markowich, G. Toscani, A. Unterreiter,*Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities*, Monatsh. Math.**133**, 1-82 (2001). MR**2002j:35188****3.**J. A. Carrillo, A. Jüngel, S. Tang,*Positive entropy schemes for a nonlinear fourth-order parabolic equation*, Discrete Cont. Dyn. Systems-B**1**, 1-20 (2003).**4.**J. A. Carrillo, G. Toscani,*Asymptotic**-decay of solutions of the porous medium equation to self-similarity*, Indiana Univ. Math. J.**49**, 113-141 (2000). MR**2001j:35155****5.**J. A. Carrillo, G. Toscani,*Long-time asymptotics for strong solutions of the thin film equation*, Comm. Math. Phys.**225**, 551-571 (2002). MR**2002m:35115****6.**B. Derrida, J. L. Lebowitz, E. Speer, H. Spohn,*Fluctuations of a stationary nonequilibrium interface*, Phys. Rev. Lett.**67**, 165-168 (1991). MR**92b:82052****7.**A. Jüngel, R. Pinnau,*Global non-negative solutions of a nonlinear fourth-oder parabolic equation for quantum systems*, SIAM J. Math. Anal.**32**, 760-777 (2000). MR**2002j:35153****8.**A. Jüngel, R. Pinnau.*A positivity preserving numerical scheme for a nonlinear fourth-order parabolic system*, SIAM J. Numer. Anal.**39**, 385-406 (2001). MR**2002h:82084****9.**A. Jüngel, G. Toscani,*Decay rates of solutions to a nonlinear fourth-order parabolic equation*, Z. Angew. Math. Phys.**54**, 377-386 (2003).**10.**J. L. Lopez, J. Soler, G. Toscani,*Time rescaling and asymptotic behavior of some fourth order degenerate diffusion equations*, Comput. Math. Appl.**43**, 721-736 (2002). MR**2002m:35127**

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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