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Long-time behavior for a nonlinear fourth-order parabolic equation
Author(s):
María
J.
Cáceres;
J.
A.
Carrillo;
G.
Toscani
Journal:
Trans. Amer. Math. Soc.
357
(2005),
1161-1175.
MSC (2000):
Primary 35K65, 35B40, 35K35
Posted:
August 11, 2004
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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.
References:
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- 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
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- 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).
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 -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49, 113-141 (2000). MR 2001j:35155 - 5.
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- 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
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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:
10.1090/S0002-9947-04-03528-7
PII:
S 0002-9947(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
Posted:
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''.
Copyright of article:
Copyright
2004,
American Mathematical Society
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