×

Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space. (English) Zbl 07461315

Summary: This paper is concerned with the large-time behavior of solutions to an initial boundary value problem for the one-dimensional bipolar Euler-Poisson equations with time-dependent damping effects \(\frac{J_i}{(1+t)^{\lambda}} (i = 1, 2)\) for \(-1 < \lambda < 1\). We first show the decay rates of the corresponding asymptotic profiles, the so-called nonlinear diffusion waves, then by means of the time-weighted energy method, we prove that the smooth solutions to the initial-boundary value problem exist uniquely and globally, and time-asymptotically converge to the nonlinear diffusion waves, provided that the initial perturbation around the diffusion wave is small enough. The convergence rates are in the forms that \(O (t^{-\frac{3}{4}(1+\lambda)})\) for \(-1 < \lambda < \frac{3}{5}\) and \(O (t^{\frac{\lambda -3}{2}})\) for \(\frac{3}{5} < \lambda < 1\), respectively, where \(\lambda = \frac{3}{5}\) is the critical point, and the convergence rate at the critical point is \(O (t^{\frac{6}{5}} \ln t)\). The results are different from those of the Cauchy problem in Li et al. (2019) [20].

MSC:

35Qxx Partial differential equations of mathematical physics and other areas of application
35Bxx Qualitative properties of solutions to partial differential equations
82Dxx Applications of statistical mechanics to specific types of physical systems
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ali, G.; Chen, L., The zero-electron-mass limit in the Euler-Poisson system for both well- and ill-prepared initial data, Nonlinearity, 24, 2745-2761 (2011) · Zbl 1227.35039
[2] Ali, G.; Jüngel, A., Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma, J. Differ. Equ., 190, 663-685 (2003) · Zbl 1020.35072
[3] Bløtekjær, K., Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17, 38-47 (1970)
[4] Cui, H.-B.; Yin, H.-Y.; Zhu, C.-J.; Zhu, L.-M., Convergence to diffusion waves for solutions of Euler equations with time-dependent damping on quadrant, Sci. China Math., 62, 33-62 (2019) · Zbl 1412.35238
[5] Donatelli, D.; Mei, M.; Rubino, B.; Sampalmieri, R., Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differ. Equ., 255, 3150-3184 (2013) · Zbl 1320.35065
[6] Gasser, I.; Hsiao, L.; Li, H.-L., Asymptotic behavior of solutions of the bipolar hydrodynamic fluids, J. Differ. Equ., 192, 326-359 (2003)
[7] Gasser, I.; Marcati, P., The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors, Math. Methods Appl. Sci., 24, 81-92 (2001) · Zbl 0974.35119
[8] Huang, F.-M.; Li, Y.-P., Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Discrete Contin. Dyn. Syst., Ser. A, 24, 455-470 (2009) · Zbl 1242.35050
[9] Huang, F.-M.; Mei, M.; Wang, Y., Large time behavior of solution to n-dimensional bipolar hydrodynamic model for semiconductors, SIAM J. Math. Anal., 43, 1595-1630 (2011) · Zbl 1228.35053
[10] Huang, F.-M.; Mei, M.; Wang, Y.; Yang, T., Long-time behavior of solution to the bipolar hydrodynamic model of semiconductors with boundary effect, SIAM J. Math. Anal., 44, 134-1164 (2012) · Zbl 1248.35020
[11] Hsiao, L.; Zhang, K.-J., The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Methods Appl. Sci., 10, 1333-1361 (2000) · Zbl 1174.82350
[12] Hsiao, L.; Zhang, K.-J., The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differ. Equ., 165, 315-354 (2000) · Zbl 0970.35150
[13] Jüngel, A., Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations (2001), Birkhäuser · Zbl 0969.35001
[14] Ju, Q.-C., Global smooth solutions to the multidimensional hydrodynamic model for plasmas with insulating boundary conditions, J. Math. Anal. Appl., 336, 888-904 (2007) · Zbl 1121.35019
[15] Ju, Q.-C.; Li, H.-L.; Li, Y.; Jiang, S., Quasi-neutral limit of the two-fluid Euler-Poisson system, Commun. Pure Appl. Anal., 9, 1577-1590 (2010) · Zbl 1213.35051
[16] Kong, H.-Y.; Li, Y.-P., Relaxation limit of the one-dimensional bipolar Euler-Poisson system in the bound domain, Appl. Math. Comput., 274, 1-13 (2016) · Zbl 1410.35104
[17] Lattanzio, C., On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10, 351-360 (2000) · Zbl 1012.82026
[18] Li, J.; Yu, H.-M., Large time behavior of solutions to a bipolar hydrodynamic model with big data and vacuum, Nonlinear Anal., Real World Appl., 34, 446-458 (2017) · Zbl 1354.35154
[19] Li, H.-T., Large Time Behavior of Solutions to Hyperbolic Equations with Time-Dependent Damping (2019), Northeast Normal University, (in Chinese), Ph.D thesis
[20] Li, H.-T.; Li, J.-Y.; Mei, M.; Zhang, K.-J., Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping, J. Math. Anal. Appl., 437, 1081-1121 (2019) · Zbl 1414.35031
[21] Li, Y.-P., Existence and some limit analysis of stationary solutions for a multi-dimensional bipolar Euler-Poisson system, Discrete Contin. Dyn. Syst., Ser. B, 16, 345-360 (2011) · Zbl 1227.35070
[22] Li, Y.-P., Global existence and asymptotic behavior of smooth solutions to a bipolar Euler-Poisson equations in a bound domain, Z. Angew. Math. Phys., 64, 1125-1144 (2013) · Zbl 1272.35037
[23] Li, Y.-P.; Liao, J., Global existence and \(L^p\) convergence rates of planar waves for three-dimensional bipolar Euler-Poisson systems, Commun. Pure Appl. Anal., 18, 1281-1302 (2019) · Zbl 1414.35171
[24] Li, Y.-P.; Yang, X.-F., Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differ. Equ., 252, 768-791 (2012) · Zbl 1242.35183
[25] Li, Y.-P.; Yang, X.-F., Pointwise estimates and \(L^p\) convergence rates to diffusion waves for a one-dimensional bipolar hydrodynamic model, Nonlinear Anal., Real World Appl., 45, 472-490 (2019) · Zbl 1409.35041
[26] Li, Y.-P.; Zhang, T., Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space, J. Differ. Equ., 251, 3143-3162 (2011) · Zbl 1228.35238
[27] Luan, L.-P.; Mei, M.; Rubino, B.; Zhu, P.-C., Large-time behavior of solutions to Cauchy problem for bipolar Euler-Poisson system with time-dependent damping in critical case, Commun. Math. Sci., 19, 1207-1231 (2021) · Zbl 1479.35110
[28] Markowich, P. A.; Ringhofev, C. A.; Schmeiser, C., Semiconductor Equations (1990), Springer-Verlag: Springer-Verlag New York, Wien
[29] Natalini, R., The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198, 262-281 (1996) · Zbl 0889.35109
[30] Peng, Y.-J.; Xu, J., Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differ. Equ., 255, 3447-3471 (2013) · Zbl 1325.35170
[31] Schochet, S., The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Commun. Math. Phys., 104, 49-75 (1986) · Zbl 0612.76082
[32] Sitnko, A.; Malnev, V., Plasma Physics Theory (1995), Chapman & Hall: Chapman & Hall London · Zbl 0845.76099
[33] Tsuge, N., Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors, Nonlinear Anal. TMA, 73, 779-787 (2010) · Zbl 1195.34044
[34] Wu, Z.-G.; Li, Y.-P., Pointwise estimates of solutions for the multi-dimensional bipolar Euler-Poisson system, Z. Angew. Math. Phys., 67, 1-20 (2016)
[35] Wu, Q.-W.; Luan, L.-P., Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping, Commun. Pure Appl. Anal., 20, 995-1023 (2021) · Zbl 1471.35242
[36] Wu, Q.-W.; Zheng, J.-Z.; Luan, L.-P., Large-time behavior of solutions to the bipolar hydrodynamic model of semiconductors with time-dependent damping, Appl. Anal. (2022), in press
[37] Zhu, C.; Hattori, H., Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differ. Equ., 166, 1-32 (2000) · Zbl 0974.35123
[38] Zhou, F.; Li, Y.-P., Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system, J. Math. Anal. Appl., 351, 480-490 (2009) · Zbl 1160.35352
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.