Large-time behavior of solutions to three-dimensional bipolar Euler–Poisson equations with time-dependent damping

Abstract

In this paper, we shall investigate the large-time behavior of solutions to the Cauchy problem for the three-dimensional bipolar isentropic Euler–Poisson equations with time-dependent damping effects $-\frac{\mu }{(1+t{)}^{\lambda }}{n}_i{u}_i(i=1,2)$ for $-1<\lambda <1,\mu >0$. Here, we consider a more general case that the two pressure functions are different and the doping profile is non-zero. Under the assumption that the initial data are close to the constant equilibrium states, we show that the smooth solutions to the Cauchy problem exist uniquely and globally. The algebraic time-decay rates of the solutions toward the constant equilibrium states are also obtained. The main ingredient of the proof is the time-weighted energy method with artfully chosen time-weighted functions.

Keywords:

• Large-time behavior
• bipolar Euler–Poisson equations
• time-dependent damping
• Cauchy problem
• time-weighted energy method

AMS Subject Classifications:

• 35Q35
• 35B40
• 76W05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported in part by the Science and Technology Commission of Shanghai Municipality [grant number 20JC1413600].