Existence, semiclassical limit and long-time behavior of weak solution to quantum drift-diffusion model

The existence, semiclassical limit and long-time behavior of weak solutions to the transient quantum drift-diffusion model are studied. Using semi-discretization in time and entropy estimate, we get the global existence and semiclassical limit of nonnegative weak solutions to one-dimensional isentropic model with nonnegative initial and homogeneous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, we obtain an inequality of the periodic weak solution to this model (or its isothermal case) which shows that the solution exponentially approaches its mean value as time increases to infinity.