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

J. Dolbeault·I. Gentil·A. Jungel
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Abstract

A nonlinear 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 is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives.

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J. Dolbeault, I. Gentil, A. Jungel. 2004-09-15. A nonlinear fourth-order parabolic equation and related logarithmic Sobolev inequalities. https://arxiv.org/abs/math/0409249

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