Asymptotic Behavior And Classical Limit Of Solutions For Bipolar Quantum Hydrodynamic Semiconductor Model

In this paper, we discuss two models. we consider a kind of one-dimensional bipolar isentropic quantum hydrodynamical model from semiconductor devices. In the first model, we show the global existence and asymptotic behavior of smooth solutions when the Plank constant ε = 1.In the second model, we discuss the classical limits of stationary solution and non-stationary solution in the case of Plank constant ε â†'0.For the first model, at first, we proves the existence of the stationary solutions by LeraySchauder fixed-point theorem. Next, we discuss the global existence and the asymptotic behavior of the smooth solutions to the initial boundary problem for a one-dimensional case in a bounded domain. The result is shown by deriving the a priori estimate uniformly in time. Here an energy form plays an essential role.For the second model, an one-dimensional bipolar isentropic quantum hydrodynamical model from semiconductor devices, first, we discuss the classical limit of the stationary solution. Then we discuss the classical limit of the non-stationary initial boundary problem for a one-dimensional case in a bounded domain. Namely, we show that the solutions to the quantum hydrodynamic model of semiconductors approaches that to the hydrodynamic model of semiconductors as the scaled Planck constants ε tends to zero.