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

María J. Cáceres; J. A. Carrillo; G. Toscani

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

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

María J. Cáceres, J. A. Carrillo, G. Toscani

Department of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

36 Citations (Scopus)

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. © 2004 American Mathematical Society.

Original language	English
Pages (from-to)	1161-1175
Journal	Transactions of the American Mathematical Society
Volume	357
DOIs	https://doi.org/10.1090/S0002-9947-04-03528-7
Publication status	Published - 1 Mar 2005

Keywords

Asymptotic behavior
Degenerate parabolic equation
Diffusion equation
Entropy dissipation

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

Cite this

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title = "Long-time behavior for a nonlinear fourth-order parabolic equation",

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. {\textcopyright} 2004 American Mathematical Society.",

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AU - Carrillo, J. A.

AU - Toscani, G.

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N2 - 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. © 2004 American Mathematical Society.

AB - 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. © 2004 American Mathematical Society.

KW - Asymptotic behavior

KW - Degenerate parabolic equation

KW - Diffusion equation

KW - Entropy dissipation

U2 - https://doi.org/10.1090/S0002-9947-04-03528-7

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VL - 357

SP - 1161

EP - 1175

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

ER -

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

Abstract

Keywords

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