Optimal convergence rate to nonlinear diffusion waves for bipolar Euler–Poisson equations with critical overdamping

Abstract

This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are L² sense by using the technical time-weighted energy method.

Keywords:

• Euler–Poisson equations
• critical over-damping
• convergence rate
• energy method

AMS (2010) Subject Classifications:

• 35L60
• 35Q35
• 35B40
• 35C06

Acknowledgments

The author would like to express her sincere thanks to Prof. Jingyu Li, Prof. Ming Mei and Prof. Kaijun Zhang for their valuable help and discussion.

Disclosure statement

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