## Asymptotic stability of a stationary solution for the bipolar full Euler-Poisson equation in a bounded domain.(English)Zbl 07446148

Summary: In this paper, we present a bipolar full hydrodynamic model from semiconductor devices, which takes the form of bipolar full Euler-Poisson with electric field and relaxation terms added to the momentum equations and energy equations. We firstly prove the existence of the stationary solutions under proper boundary value conditions. Next, we present the global existence and the asymptotic behavior of smooth solutions to the initial boundary value problem for the one-dimensional bipolar full Euler-Poisson system in a bounded domain. The result is shown by an elementary energy method. Compared with the corresponding initial data case, we find that the asymptotic state of the solutions is the stationary solution.

### 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
Full Text:

### References:

 [1] Anile, A. M.; Muscato, O., Improved hydrodynamical model from carrier transport in semiconductors, Phys. Rev. B, 51, 16728-16740 (1995) [2] Anile, A. M.; Pennisi, S., Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors, Cotinuum Mech. Thermodyn., 41, 187-197 (1992) [3] Chen, D. P.; Eisenberg, R. S.; Jerome, J. W.; Shu, C. W., A hydrodynamic model of temperature change in open ionic channnels, Biophys. J., 69, 2304-2322 (1995) [4] Blotekjaer, K., Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron. Dev., ED-17, 38-47 (1970) [5] Jüngel, A., Quasi-hydrodynamic semiconductor equations, (Progress in Nonlinear Differential Equations (2001), Birkhäuser) · Zbl 0969.35001 [6] Markowich, P. A.; Ringhofev, C. A.; Schmeiser, C., Semiconductor Equations (1990), Springer-Verlag, Wien: Springer-Verlag, Wien New York [7] Sitnko, A.; Malnev, V., Plasma Physics Theory (1995), Chapman & Hall: Chapman & Hall London · Zbl 0845.76099 [8] 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 [9] 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 [10] Natalini, R., The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198, 262-281 (1996) · Zbl 0889.35109 [11] Hsiao, L.; Zhang, K.-J., The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations, J. Differential Equations, 165, 315-354 (2000) · Zbl 0970.35150 [12] 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 [13] 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 [14] 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 [15] 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 [16] Zhu, C.; Hattori, H., Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166, 1-32 (2000) · Zbl 0974.35123 [17] Gasser, I.; Hsiao, L.; Li, H.-L., Asmptotic behavior of solutions of the bipolar hydrodynamic fluids, J. Differential Equations, 192, 326-359 (2003) [18] 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 [19] Li, J.; Yu, H.-M., Large time behavior of solutions to a bipolar hydrodynamic model with big data and vacuum, Nonlinear Analysis RWA, 34, 446-458 (2017) · Zbl 1354.35154 [20] 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 [21] Li, Y.-P.; Yang, X.-F., Pointwise estimates and $$L^p$$ convergence rates to diffusion waves for a one-dimensional bipolar hydrodynamic model, Nonlinear Analysis RWA, 45, 472-490 (2019) · Zbl 1409.35041 [22] 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 [23] Donatelli, D.; Mei, M.; Rubino, B.; Sampalmieri, R., Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255, 3150-3184 (2013) · Zbl 1320.35065 [24] 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 [25] Ali, G.; Jüngel, A., Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma, J. Differential Equations, 190, 663-685 (2003) · Zbl 1020.35072 [26] 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 [27] 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 [28] Li, Y.-P., Diffusion relaxation limit of a bipolar isentropic hydrodynamic model for semiconductors, J. Math. Anal. Appl., 336, 1341-1356 (2007) · Zbl 1121.35011 [29] 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 [30] 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 [31] Li, Y.-P.; Yang, X.-F., Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252, 768-791 (2012) · Zbl 1242.35183 [32] Li, Y.-P.; Zhang, T., Relaxation-time limit of the multidimensional bipolar hydrodynamic model in Besov space, J. Differential Equations, 251, 3143-3162 (2011) · Zbl 1228.35238 [33] Peng, Y.-J.; Xu, J., Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255, 3447-3471 (2013) · Zbl 1325.35170 [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] Li, Y.-P., Global existence and asymptotic behavior of solutions to the nonisentropic bipolar hydrodynamic models, J. Differential Equations, 250, 1285-1309 (2011) · Zbl 1227.35069 [36] Hu, H.-F.; Zhang, K.-J., Stability of the stationary solution of the Cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate, Kinet. Relat. Models, 8, 117-151 (2015) · Zbl 1332.35035 [37] Li, Y.-P., Global existence and asymptotic behavior of the solutions to the 3D bipolar non-isentropic Euler-Poisson equation, Nonlinear Anal. Model. Control, 20, 305-330 (2015) · Zbl 1421.35265 [38] Jiang, S.; Ju, Q.-C.; Li, H.-L.; Li, Y., Quasi-neutral limit of the full bipolar Euler-Poisson system, Sci. China Math., 53, 3099-3114 (2010) · Zbl 1215.35022 [39] Li, Y.-P.; Zhou, Z.-M., Relaxation-time limit in the multi-dimensional bipolar nonisentropic Euler-Poisson systems, J. Differential Equations, 258, 3546-3566 (2015) · Zbl 1311.35011 [40] Li, Y.-P.; Lu, L., Stability of planar diffusion wave for the three dimensional full bipolar Euler-Poisson system, Appl. Math. Comput., 356, 392-410 (2019) · Zbl 1428.35381 [41] Nishibata, S.; Suzuki, M., Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44, 639-665 (2007) · Zbl 1138.82033 [42] Nishibata, S.; Suzuki, M., Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 244, 836-874 (2008) · Zbl 1139.82042 [43] Schochet, S., The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Comm. Math. Phys., 104, 49-75 (1986) · Zbl 0612.76082
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.