Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space

Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping

“…Luan, et al [25] studied the Cauchy problem for critical under-damping case λ = 1, µ > 2. Wu and Li [32] investigated the initial boundary value problem for −1 < λ < 1, µ > 0. For the bipolar quantum Euler-Poisson system with time-dependent damping, Wu and Li [31] proved the global existence of smooth solutions to the Cauchy problem and showed the optimal converge rates of the smooth solutions toward the nonlinear diffusion waves.…”

“…But in the critical over-damping case λ = −1, the electric field E is no longer the time-exponentially decaying and is just time-algebraically decay, which leads us cannot ignore the effect of the electric field. Next, due to the decay rates of the nonlinear diffusion waves are in logarithmic form, the method that constructing the polynomial time weights used in [18,32,31] is no longer applicable in this case. To overcome this difficulty, inspired by [19], we technically construct logarithmic time weights when establishing the a-priori estimates.…”

“…Moreover, from the a-priori assumptions (33) and the Sobolev inequality as well as the relation (32), we have for i = 1, 2…”

Asymptotic stability of nonlinear diffusion waves for the bipolar quantum Euler–Poisson system with time‐dependent damping

“…Furthermore, Wu et al 26 investigated a more physical and a challenging case that two pressure functions can be different and with a nonzero doping profile. Later, Wu et al 27 studied the stability of the nonlinear diffusion wave in a half line. In this paper, we study the Cauchy problem for the system (1.1) with the initial data and the far field conditions…”

“…Furthermore, Wu et al 26 investigated a more physical and a challenging case that two pressure functions can be different and with a nonzero doping profile. Later, Wu et al 27 studied the stability of the nonlinear diffusion wave in a half line. In this paper, we study the Cauchy problem for the system () with the initial data and the far field conditions $$$$ \left\{\begin{array}{c}\left({n}_1,{n}_2,{J}_1,{J}_2\right)\left(x,0\right)&#x0003D;\left({n}_{10},{n}_{20},{J}_{10},{J}_{20}\right)(x)\to \left({n}_{\pm },{n}_{\pm },{J}_{1\pm },{J}_{2\pm}\right)\kern1em \mathrm{as}\kern1em x\to \pm \infty, \\ {}E\left(-\infty, t\right)&#x0003D;{E}&#x0005E;{-},\end{array}\right.$$…”