The Existence Of One-dimensional Quantum Monopoles And Big-time Behavior Equations Euler-Poisson

In this paper, I study the one-dimensional isentropic quantum hydrodynamic system forsemiconductors. It takes the isentropic Euler-Poisson equations with the quantum potentialand momentum relaxation term in the momentum equations. I show the global existenceand large-time behavior of smooth solutions for the initial value problem to one-dimensionalquantum Euler-Poisson equations, when the far field states of current density are inconsistentand the far field of the electric field is not zero. Choosing proper corrections and by the energymethods, we show that smooth solutions of the initial problem globally existed and decayexponentially fast to the corresponding stationary solutions. This result improve the previousresults in which the current density’s for fields are equal and the far field of the electric field is zero.