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A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities

Dolbeault, Jean; Gentil, Ivan; Jüngel, Ansgar (2006), A logarithmic fourth-order parabolic equation and related logarithmic Sobolev inequalities, Communications in Mathematical Sciences, 4, 2, p. 275-290

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Type
Article accepté pour publication ou publié
Date
2006
Journal name
Communications in Mathematical Sciences
Volume
4
Number
2
Publisher
International Press
Pages
275-290
Metadata
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Author(s)
Dolbeault, Jean cc
Gentil, Ivan
Jüngel, Ansgar
Abstract (EN)
A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions and some regularity results are shown. Furthermore, we prove that the solution converges exponentially fast to its mean value in the "entropy norm" and in the Fisher information, using a new optimal logarithmic Sobolev inequality for higher derivatives. In particular, the rate is independent of the solution and the constant depends only on the initial value of the entropy.
Subjects / Keywords
Cauchy problem; higher-order parabolic equations; existence of global-in-time solutions; long-time behavior; Fisher information; entropy{entropy production method; logarithmic Sobolev inequality; Poincaré inequality