Asymptotic Convergence to Stationary Waves for Unipolar Hydrodynamic Model of Semiconductors

In this paper, we study the one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler–Poisson equations. In the case when the state constants on the current density and the electric field are nonzero (switch-on case), the stability of stationary waves of one-dimensional isentropic Euler–Poisson equations for the unipolar hydrodynamic model has been open. In order to overcome this difficulty, we first analyze the behaviors of the solutions at $x=\pm\infty$, and observe what are the exact gaps between the original solutions and the stationary solutions in $L^2$-space; then we technically construct some new correction functions to delete these gaps. Finally, based on the energy methods, we prove that the solutions of one-dimensional isentropic Euler–Poisson equations for the unipolar hydrodynamic model decay exponentially fast to the stationary solutions.