Asymptotic behavior of solutions to bipolar Euler–Poisson equations with time-dependent damping

Abstract

In this paper, we study the one-dimensional Euler–Poisson equations of bipolar hydrodynamic model for semiconductor device with time-dependent damping effect − J ( 1 + t ) λ for − 1 < λ < 1 , where the damping effect is time-gradually-degenerate for λ > 0 , and time-gradually-enhancing for λ < 0 . Such a damping effect makes the hydrodynamic system possess the nonlinear diffusion phenomena time-asymptotic-weakly or strongly. Based on technical observation, and by using the time-weighted energy method, where the weights are artfully chosen, we prove that the system admits a unique global smooth solution, which time-asymptotically converges to the corresponding diffusion wave, when the initial perturbation around the diffusion wave is small enough. The convergence rates are specified in the algebraic forms O ( t − 3 4 ( 1 + λ ) ) and O ( t − ( 1 − λ ) ) according to different values of λ in ( − 1 , 1 7 ) and ( 1 7 , 1 ) , respectively, where λ = 1 7 is the critical point, and the convergence rate at the critical point is O ( t − 6 7 ln ⁡ t ) . All these convergence rates obtained in different cases are optimal in the sense when the initial perturbations are L 2 -integrable. Particularly, when λ = 1 7 , the convergence rate is the fastest, namely, the asymptotic profile of the original system at the critical point is the best.