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# Communications in Mathematical Sciences

## Volume 20 (2022)

### Number 2

### Sub-exponential convergence to steady-states for 1-D Euler–Poisson equations with time-dependent damping

Pages: 447 – 478

DOI: https://dx.doi.org/10.4310/CMS.2022.v20.n2.a7

#### Authors

#### Abstract

This paper is concerned with the Cauchy problem to Euler–Poisson equations for 1‑D unipolar hydrodynamic model of semiconductors with time-dependent damping effect $-\frac{nu}{(1+t)^\lambda}$ for $\lambda \in (-1,0) \cup (0,1)$, where the damping is strong for $ \lambda \lt 0$ and weak for $ \lambda \gt 0$. For the strong damping case with $\lambda \in (-1,0)$, the system is proved to possess a unique global smooth solution time-asymptotically converging to the steady-state in the sub-exponential form $O((1+t)^{\lvert\lambda\rvert} e^{-\alpha(1+t)^{1-\lvert\lambda\rvert}})$ for some constant $\alpha \gt 0$. For the weak damping case with $\lambda \in (0,1)$, when the doping profile is completely flat, the system is further proved to admit a unique global smooth solution converging to the constant steady-state in the sub-exponential form $O((1+t){-\frac{\lvert \theta+\lambda \rvert}{2}} e-{\beta(1+t)^{1-\lvert\lambda\rvert}})$ for some number $\beta \gt 0$. Specially, the index $ \theta\in[\lambda,\infty)$ relies on the initial perturbation and could be large enough once the initial perturbation is sufficiently close to zero, such that the convergence rate involving the part of algebraic decay can be arbitrarily large. A new observation is that the time-dependent damping essentially affects the asymptotic behavior of solutions to Euler–Poisson system, and both the weak and strong damping effects cause the decay rates to be sub-exponential, which are slower than the regular exponential decay in the case of $\lambda=0$. The adopted approach for the proof in this paper is based on the elementary $L^2$-energy estimates but with some technical development.

#### Keywords

Euler–Poisson equations, unipolar hydrodynamic model, semiconductor, weak damping, strong damping, sub-exponential convergence, steady-states

#### 2010 Mathematics Subject Classification

35B40, 35L50, 35L60, 35L65

Received 25 January 2021

Received revised 11 June 2021

Accepted 27 July 2021

Published 28 January 2022