Decay estimates and large time behavior of solutions to the Drift-Diffusion system

We study the drift-diffusion system arising in the semiconductor device simulation and the plasma physics. We consider the initial value problem for this system and derive the optimal L^p decay estimate of solutions by applying the time weighted L^p energy method. Furthermore, we show that the solutions approach the corresponding heat kernels as time tends to infinity. This asymptotic result is based on the optimal L^p -L^q decay estimate for the linearized problem.