doi: 10.3934/dcdsb.2022008
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Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping

Department of Mathematics, Shanghai University, Shanghai 200444, China

*Corresponding author: Qiwei Wu

Received  September 2021 Early access January 2022

Fund Project: The author is supported in part by Science and Technology Commission of Shanghai Municipality (Grant No. 20JC1413600)

We shall investigate the large-time behavior of solutions to the Cauchy problem for the one-dimensional bipolar quantum Euler-Poisson system with critical time-dependent over-damping. By means of the time-weighted energy method, we prove that the smooth solutions to the Cauchy problem exist uniquely and globally, and time-asymptotically converge to the nonlinear diffusion waves when the initial perturbation around the nonlinear diffusion waves are small enough. Particularly, we show the optimal decay rates of solutions toward the nonlinear diffusion waves.

Citation: Qiwei Wu. Large-time behavior of solutions to the bipolar quantum Euler-Poisson system with critical time-dependent over-damping. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022008
References:
[1]

S. ChenH. LiM. Mei and K. Zhang, Global and blow-up solutions to compressible Euler equations with time-dependent damping, J. Differential Equations, 268 (2020), 5035-5077.  doi: 10.1016/j.jde.2019.11.002.

[2]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[3]

D. DonatelliM. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255 (2013), 3150-3184.  doi: 10.1016/j.jde.2013.07.027.

[4]

C. J. Peletier and L. A. van Duyn amd, A class of similary solutions of the nonlinear diffusion equations, Nonlinear Anal, 1 (1977), 223-233.  doi: 10.1016/0362-546X(77)90032-3.

[5]

D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.

[6]

C. L. Gardner, The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math., 54 (1994), 409-427.  doi: 10.1137/S0036139992240425.

[7]

I. GasserL. Hsiao and H. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.

[8]

I. Gasser and P. A. Markowich, Quantum hydrodunamics, Wigner transforms and the classical limit, Asymptot. Anal., 14 (1997), 97-116.  doi: 10.3233/ASY-1997-14201.

[9]

S. GengY. Lin and M. Mei, Asymptotic behavior of solutions to Euler equations with time-dependent damping in critical case, SIAM J. Math. Anal., 52 (2020), 1463-1488.  doi: 10.1137/19M1272846.

[10]

H. HuM. Mei and K. Zhang, Asymptotic stability and semi-classical limit for bipolar quantum hydrodynamic model, Commun. Math. Sci., 14 (2016), 2331-2371.  doi: 10.4310/CMS.2016.v14.n8.a10.

[11]

J. HuY. Li and J. Liao, The stationary solution of a one-dimensional bipolar quantum hydrodynamic model, J. Math. Anal. Appl., 493 (2021), 124537.  doi: 10.1016/j.jmaa.2020.124537.

[12]

F. HuangH.-L. Li and A. Matsumura, Existence and stability of steady-state of one-dimensional quantum Euler-Poisson system for semiconductors, J. Differential Equations, 225 (2006), 1-25.  doi: 10.1016/j.jde.2006.02.002.

[13]

F. Huang and Y. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Dis. Contin. Dyn. Sys., Ser. A, 24 (2009), 455-470.  doi: 10.3934/dcds.2009.24.455.

[14]

F. HuangM. Mei and Y. Wang, Large time behavior of solutions to n-dimensional bipolar hydrodynamic models for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630.  doi: 10.1137/100810228.

[15]

F. HuangM. MeiY. Wang and T. Yang, Long-time behavior of solution to the bipolar hydrodynamic model of semiconductors with boundary effect, SIAM J. Math. Anal., 44 (2012), 1134-1164.  doi: 10.1137/110831647.

[16]

N. C. KlusdahlA. M. KrimanD. K. Ferry and C. Ringhofer, Self-consistent study of the resonant-tunneling diode, Phis. Rev. B, 39 (1989), 7720-7735.  doi: 10.1103/PhysRevB.39.7720.

[17]

C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360.  doi: 10.1142/S0218202500000215.

[18]

H. LiJ. LiM. Mei and K. Zhang, Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping, J. Math. Anal. Appl., 437 (2019), 1081-1121.  doi: 10.1016/j.jmaa.2019.01.010.

[19]

H. LiJ. LiM. Mei and K. Zhang, Optimal convergence rate to nonlinear diffusion waves for Euler equations with cirical overdamping, Appl. Math. Lett., 113 (2021), 106882.  doi: 10.1016/j.aml.2020.106882.

[20]

H.-L. LiG. Zhang and K. Zhang, Algebraic time-decay for the bipolar quantum hydrodynamic model, Math. Models Methods Appl. Sci., 18 (2008), 859-881.  doi: 10.1142/S0218202508002887.

[21]

Y. Li, Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models, Nonlinear Anal., 74 (2011), 1501-1512.  doi: 10.1016/j.na.2010.10.023.

[22]

Y. Li, Global existence and large time behavior of solutions for the bipolar quantum hydrodynamic models in the quarter plane, Math. Meth. Appl. Sci., 36 (2013), 1409-1422.  doi: 10.1002/mma.2694.

[23]

Y. Li and X. Yang, Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791.  doi: 10.1016/j.jde.2011.08.008.

[24]

B. Liang and K. Zhang, Steady-state solutions and asymptotic limits on the multi-dimensional semiconductor quantum hydrodynamic model, Math. Models Methods Appl. Sci., 17 (2007), 253-275.  doi: 10.1142/S0218202507001905.

[25]

L. LuanM. MeiB. Rubino and P. Zhu, Large-time behavior of solutions to Cauchy problem for bipolar Euler-Poisson system with time-dependent damping in critical case, Commun. Math. Sci., 19 (2021), 1207-1231.  doi: 10.4310/CMS.2021.v19.n5.a2.

[26]

M. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded damain, Kinet. Relat. Models, 5 (2012), 537-550.  doi: 10.3934/krm.2012.5.537.

[27]

S. Nishibata and M. Suzuki, Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits, J. Differential Equations, 244 (2008), 836-874.  doi: 10.1016/j.jde.2007.10.035.

[28]

X. Pan, Global existence of solutions to 1-d Euler equations with time-dependent damping, Nonlinear Anal., 132 (2016), 327-336.  doi: 10.1016/j.na.2015.11.022.

[29]

X. Pan, Blow up of solutions to 1-d Euler equations with time-dependent damping, J. Math. Anal. Appl., 442 (2016), 435-445.  doi: 10.1016/j.jmaa.2016.04.075.

[30]

A. Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Commun. Math. Phys., 188 (1997), 69-88.  doi: 10.1007/s002200050157.

[31]

Q.-W. Wu and Y.-P. Li, Asymptotic behavior of solutions to the bipolar quantum Euler-Poisson system with time-dependent damping, preprint, 2021.

[32]

Q. WuY. Li and R. Xu, Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space, J. Math. Anal. Appl., 508 (2022), 125899.  doi: 10.1016/j.jmaa.2021.125899.

[33]

Q. Wu, J. Zheng and L. Luan, Large-time behavior of solutions to the time-dependent damper bipolar Euler-Poisson system, Appl. Anal., (2021), in press. doi: 10.1080/00036811.2021.1969015.

[34]

G. ZhangH.-L. Li and K. Zhang, Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors, J. Differential Equations, 245 (2008), 1433-1453.  doi: 10.1016/j.jde.2008.06.019.

[35]

G. Zhang and K. Zhang, On the bipolar quantum Euler-Poisson system: The thermal equilibrium model solution and semiclassical limit, Nonlinear Anal., 66 (2007), 2218-2229.  doi: 10.1016/j.na.2006.03.010.

show all references

References:
[1]

S. ChenH. LiM. Mei and K. Zhang, Global and blow-up solutions to compressible Euler equations with time-dependent damping, J. Differential Equations, 268 (2020), 5035-5077.  doi: 10.1016/j.jde.2019.11.002.

[2]

P. Degond and C. Ringhofer, Quantum moment hydrodynamics and the entropy principle, J. Stat. Phys., 112 (2003), 587-628.  doi: 10.1023/A:1023824008525.

[3]

D. DonatelliM. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differential Equations, 255 (2013), 3150-3184.  doi: 10.1016/j.jde.2013.07.027.

[4]

C. J. Peletier and L. A. van Duyn amd, A class of similary solutions of the nonlinear diffusion equations, Nonlinear Anal, 1 (1977), 223-233.  doi: 10.1016/0362-546X(77)90032-3.

[5]

D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950.  doi: 10.1103/PhysRevB.48.7944.

[6]

C. L. Gardner, The quantum hydrodynamic model for semiconductors devices, SIAM J. Appl. Math., 54 (1994), 409-427.  doi: 10.1137/S0036139992240425.

[7]

I. GasserL. Hsiao and H. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.

[8]

I. Gasser and P. A. Markowich, Quantum hydrodunamics, Wigner transforms and the classical limit, Asymptot. Anal., 14 (1997), 97-116.  doi: 10.3233/ASY-1997-14201.

[9]

S. GengY. Lin and M. Mei, Asymptotic behavior of solutions to Euler equations with time-dependent damping in critical case, SIAM J. Math. Anal., 52 (2020), 1463-1488.  doi: 10.1137/19M1272846.

[10]

H. HuM. Mei and K. Zhang, Asymptotic stability and semi-classical limit for bipolar quantum hydrodynamic model, Commun. Math. Sci., 14 (2016), 2331-2371.  doi: 10.4310/CMS.2016.v14.n8.a10.

[11]

J. HuY. Li and J. Liao, The stationary solution of a one-dimensional bipolar quantum hydrodynamic model, J. Math. Anal. Appl., 493 (2021), 124537.  doi: 10.1016/j.jmaa.2020.124537.

[12]

F. HuangH.-L. Li and A. Matsumura, Existence and stability of steady-state of one-dimensional quantum Euler-Poisson system for semiconductors, J. Differential Equations, 225 (2006), 1-25.  doi: 10.1016/j.jde.2006.02.002.

[13]

F. Huang and Y. Li, Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum, Dis. Contin. Dyn. Sys., Ser. A, 24 (2009), 455-470.  doi: 10.3934/dcds.2009.24.455.

[14]

F. HuangM. Mei and Y. Wang, Large time behavior of solutions to n-dimensional bipolar hydrodynamic models for semiconductors, SIAM J. Math. Anal., 43 (2011), 1595-1630.  doi: 10.1137/100810228.

[15]

F. HuangM. MeiY. Wang and T. Yang, Long-time behavior of solution to the bipolar hydrodynamic model of semiconductors with boundary effect, SIAM J. Math. Anal., 44 (2012), 1134-1164.  doi: 10.1137/110831647.

[16]

N. C. KlusdahlA. M. KrimanD. K. Ferry and C. Ringhofer, Self-consistent study of the resonant-tunneling diode, Phis. Rev. B, 39 (1989), 7720-7735.  doi: 10.1103/PhysRevB.39.7720.

[17]

C. Lattanzio, On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit, Math. Models Methods Appl. Sci., 10 (2000), 351-360.  doi: 10.1142/S0218202500000215.

[18]

H. LiJ. LiM. Mei and K. Zhang, Asymptotic behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping, J. Math. Anal. Appl., 437 (2019), 1081-1121.  doi: 10.1016/j.jmaa.2019.01.010.

[19]

H. LiJ. LiM. Mei and K. Zhang, Optimal convergence rate to nonlinear diffusion waves for Euler equations with cirical overdamping, Appl. Math. Lett., 113 (2021), 106882.  doi: 10.1016/j.aml.2020.106882.

[20]

H.-L. LiG. Zhang and K. Zhang, Algebraic time-decay for the bipolar quantum hydrodynamic model, Math. Models Methods Appl. Sci., 18 (2008), 859-881.  doi: 10.1142/S0218202508002887.

[21]

Y. Li, Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models, Nonlinear Anal., 74 (2011), 1501-1512.  doi: 10.1016/j.na.2010.10.023.

[22]

Y. Li, Global existence and large time behavior of solutions for the bipolar quantum hydrodynamic models in the quarter plane, Math. Meth. Appl. Sci., 36 (2013), 1409-1422.  doi: 10.1002/mma.2694.

[23]

Y. Li and X. Yang, Global existence and asymptotic behavior of the solutions to the three dimensional bipolar Euler-Poisson systems, J. Differential Equations, 252 (2012), 768-791.  doi: 10.1016/j.jde.2011.08.008.

[24]

B. Liang and K. Zhang, Steady-state solutions and asymptotic limits on the multi-dimensional semiconductor quantum hydrodynamic model, Math. Models Methods Appl. Sci., 17 (2007), 253-275.  doi: 10.1142/S0218202507001905.

[25]

L. LuanM. MeiB. Rubino and P. Zhu, Large-time behavior of solutions to Cauchy problem for bipolar Euler-Poisson system with time-dependent damping in critical case, Commun. Math. Sci., 19 (2021), 1207-1231.  doi: 10.4310/CMS.2021.v19.n5.a2.

[26]

M. MeiB. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded damain, Kinet. Relat. Models, 5 (2012), 537-550.  doi: 10.3934/krm.2012.5.537.

[27]

S. Nishibata and M. Suzuki, Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits, J. Differential Equations, 244 (2008), 836-874.  doi: 10.1016/j.jde.2007.10.035.

[28]

X. Pan, Global existence of solutions to 1-d Euler equations with time-dependent damping, Nonlinear Anal., 132 (2016), 327-336.  doi: 10.1016/j.na.2015.11.022.

[29]

X. Pan, Blow up of solutions to 1-d Euler equations with time-dependent damping, J. Math. Anal. Appl., 442 (2016), 435-445.  doi: 10.1016/j.jmaa.2016.04.075.

[30]

A. Unterreiter, The thermal equilibrium solution of a generic bipolar quantum hydrodynamic model, Commun. Math. Phys., 188 (1997), 69-88.  doi: 10.1007/s002200050157.

[31]

Q.-W. Wu and Y.-P. Li, Asymptotic behavior of solutions to the bipolar quantum Euler-Poisson system with time-dependent damping, preprint, 2021.

[32]

Q. WuY. Li and R. Xu, Large-time behavior of solutions to bipolar Euler-Poisson equations with time-dependent damping in the half space, J. Math. Anal. Appl., 508 (2022), 125899.  doi: 10.1016/j.jmaa.2021.125899.

[33]

Q. Wu, J. Zheng and L. Luan, Large-time behavior of solutions to the time-dependent damper bipolar Euler-Poisson system, Appl. Anal., (2021), in press. doi: 10.1080/00036811.2021.1969015.

[34]

G. ZhangH.-L. Li and K. Zhang, Semiclassical and relaxation limits of bipolar quantum hydrodynamic model for semiconductors, J. Differential Equations, 245 (2008), 1433-1453.  doi: 10.1016/j.jde.2008.06.019.

[35]

G. Zhang and K. Zhang, On the bipolar quantum Euler-Poisson system: The thermal equilibrium model solution and semiclassical limit, Nonlinear Anal., 66 (2007), 2218-2229.  doi: 10.1016/j.na.2006.03.010.

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