Large-time behavior of solutions to the time-dependent damped bipolar Euler-Poisson system

Abstract

This paper concerns with the Cauchy problem of the 1-D bipolar hydrodynamic model for semiconductors, a system of Euler-Poisson equations with time-dependent damping effects $-J(1+t{)}^{-\lambda }$ and $-K(1+t{)}^{-\lambda }$ for $-1<\lambda <1$. Here, we consider a more physical case that allows the two pressure functions can be different and the doping profile can be non-zero. Different from the previous study [Li HT, Li JY, Mei. M, et al. Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping. J Math Anal Appl. 2019;437:1081-1121] which considered two identical pressure functions and zero doping profile, the asymptotic profiles of the solutions to this model are constant states rather than the nonlinear diffusion waves. When the initial perturbation around the constant states are sufficiently small in the sense of $L^2$, by means of the time-weighted energy method, we prove the global existence and uniqueness of the smooth solutions to the Cauchy problem, and obtain the optimal convergence rates of the solutions toward the constant states.

Keywords:

• Bipolar hydrodynamic model
• Euler-Poisson equations
• time-dependent damping
• large-time behavior
• time-weighted energy method

AMS:

• 35B40
• 35B65
• 35Q35

Acknowledgments

We are grateful to the two referees' valuable comments and suggestions, which led a significant improvement of our original manuscript We also wish here to express his sincere appreciation to Prof. Peicheng Zhu for his careful guidance during the preparation of the paper. Wu is supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Wu is supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600).