  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.
