Algebraic time-decay for the bipolar quantum hydrodynamic model

TL;DR

This paper proves that solutions to the bipolar quantum hydrodynamic model for semiconductors in three dimensions decay algebraically over time, highlighting differences from unipolar models and the influence of nonlinear coupling.

Contribution

It establishes the global existence and algebraic decay rates of solutions for the bipolar QHD system, revealing the impact of nonlinear coupling and dispersion on long-term behavior.

Findings

Global strong solutions exist and decay algebraically in time.

Decay rates are proven to be both upper and lower bounds.

Nonlinear dispersion does not alter the algebraic decay rate.

Abstract

The initial value problem is considered in the present paper for bipolar quantum hydrodynamic model for semiconductors (QHD) in $\mathbb{R}^3$. We prove that the unique strong solution exists globally in time and tends to the asymptotical state with an algebraic rate as $t\to+\infty$. And, we show that the global solution of linearized bipolar QHD system decays in time at an algebraic decay rate from both above and below. This means in general, we can not get exponential time-decay rate for bipolar QHD system, which is different from the case of unipolar QHD model (where global solutions tend to the equilibrium state at an exponential time-decay rate) and is mainly caused by the nonlinear coupling and cancelation between two carriers. Moreover, it is also shown that the nonlinear dispersion does not affect the long time asymptotic behavior, which by product gives rise to the algebraic time-decay rate of the solution of the bipolar hydrodynamical model in the semiclassical limit.