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# Communications in Mathematical Sciences

## Volume 4 (2006)

### Number 2

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

Pages: 275 – 290

DOI: http://dx.doi.org/10.4310/CMS.2006.v4.n2.a1

#### Authors

#### Abstract

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.