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

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.