The Well-posedness Of A Bipolar Semiconductor Quantum Model

In this thesis, I study a bipolar quantum drift-diffusion model fromsemiconductor devices and plasmas, which consists of two non-linearfourth-order parabolic equations and one Poisson equation.First, for the proper boundary conditions, using the standard theoryof elliptic equation and Schauder fixed point theorem and the carefulenergy estimates, I establish the unique existence of the stationarysolutions to the one-dimensional bipolar quantum drift-diffusionequations. I also discuss the classical limit of the stationary solutions tothe bipolar quantum drift-diffusion model equations. Namely, I showthat the stationary solution to the quantum drift-diffusion equationsapproaches that to the classical drift-diffusion equations as the scaledPlanck constant$\varepsilon$tends to zero.Next, using an iteration and Galerkin method, I discuss the uniqueexistence locally in time of the non-stationary solutions to theone-dimensional bipolar quantum drift-diffusion equations under theproper initial and boundary conditions.