Communications in Mathematical Sciences

Volume 4 (2006)

Number 2

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

Pages: 275 – 290



Jean Dolbeault

Ivan Gentil

Ansgar Jüngel


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.

2010 Mathematics Subject Classification

Primary 35K55. Secondary 35B40, 35K35.

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