American Institute of Mathematical Sciences

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March  2021, 20(3): 995-1023. doi: 10.3934/cpaa.2021003

Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping

Qiwei Wu 1, and  Liping Luan 2,,

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China
2. 

Materials Genome Institute, Shanghai University, Shanghai 200444, China

* Corresponding author

Received  August 2020 Revised  November 2020 Published  January 2021

This paper is concerned with the Cauchy problem of the 1-D unipolar hydrodynamic model for semiconductor device, a system of Euler-Poisson equations with time-dependent damping effect $ -J(1+t)^{-\lambda} $ for $ -1<\lambda<1 $, where $ J $ denotes the current density, and the damping effect is asymptotically vanishing as $ t \to \infty $ for $ \lambda>0 $, and asymptotically enhancing to infinity as $ t \to \infty $ for $ \lambda<0 $. When the initial perturbation around the constant states are sufficiently small, by means of the time-weighted energy method, we prove that the smooth solutions to the Cauchy problem exist uniquely and globally. Particularly, we also obtain the large-time behavior of the solutions.

Keywords: Unipolar hydrodynamic model, Euler-Poisson equations, time-dependent damping, large-time behavior, time-weighted energy method.
Mathematics Subject Classification: Primary: 35Q35, 35B40; Secondary: 35B65.
Citation: Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
References:
[1]

K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron. Devices, 17 (1970), 38-47.   Google Scholar

[2]

S. G. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-up solutions to compressible Euler equations with time-dependent damping, J. Differ. Equ., 268 (2020), 5035-5077.  doi: 10.1016/j.jde.2019.11.002.  Google Scholar

[3]

G. Q. Chen and D. Wang, Convergence of shock schemes for the compressible Euler-Poisson equations, Commun. Math. Phys., 179 (1996), 333-364.   Google Scholar

[4]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[5]

P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Math. Pure Appl., 4 (1993), 87-98.  doi: 10.1007/BF01765842.  Google Scholar

[6]

D. Donatelli, M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differ. Equ., 255 (2013), 3150-3184. doi: 10.1016/j.jde.2013.07.027.  Google Scholar

[7]

W. F. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differ. Equ., 133 (1997), 224-244.  doi: 10.1006/jdeq.1996.3203.  Google Scholar

[8]

I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Commun. Partial Differ. Equ., 17 (1992), 553-577.  doi: 10.1080/03605309208820853.  Google Scholar

[9]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179 (2006), 1-30.  doi: 10.1007/s00205-005-0369-2.  Google Scholar

[10]

I. Gasser, L. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differ. Equ., 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.  Google Scholar

[11]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differ. Equ., 192 (2003), 111-133.  doi: 10.1016/S0022-0396(03)00063-9.  Google Scholar

[12]

L. Hsiao and T. Yang, Asymptotic of Initial Boundary Value Problems for Hydrodynamic and Drift Diffusion Models for Semiconductors, J. Differ. Equ., 170 (2001), 473-493.  doi: 10.1006/jdeq.2000.3825.  Google Scholar

[13]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initialboundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Meth. Appl. Sci., 10 (2000), 1333-1361.  doi: 10.1142/S0218202500000653.  Google Scholar

[14]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations, J. Differ. Equ., 165 (2000), 315-354.  doi: 10.1006/jdeq.2000.3780.  Google Scholar

[15]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429.  doi: 10.1137/100793025.  Google Scholar

[16]

F. M. Huang, M. 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.  Google Scholar

[17]

F. M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44 (2012), 1134-1164.  doi: 10.1137/110831647.  Google Scholar

[18]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors, J. Differ. Equ., 251 (2011), 1305-1331.  doi: 10.1016/j.jde.2011.04.007.  Google Scholar

[19]

H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh Sect. A, 132 (2002), 359-378.  doi: 10.1017/S0308210500001670.  Google Scholar

[20]

H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic shock solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  doi: 10.1090/qam/1939010.  Google Scholar

[21]

H. T. Li, Large Time Behavior of Solutions to Hyperbolic Equations with Time-Dependent Damping (in Chinese), Ph.D thesis, Northeast Normal University, 2019. Google Scholar

[22]

H. T. Li, J. Y. Li, M. Mei and K. J. 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.  Google Scholar

[23]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830.  doi: 10.1137/S0036139996312168.  Google Scholar

[24]

L. P. Luan, M. Mei, B. Rubino and P. C. Zhu, Large-Time Behavior of Solutions to Cauchy Problem for Bipolar Euler-Poisson System with Time-Dependent Damping in Critical Case, preprint, 2020.  Google Scholar

[25]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation, Publ. RIMS Kyoto Univ., 13 (1977), 349-379.  doi: 10.2977/prims/1195189813.  Google Scholar

[27]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rat. Mech. Anal., 129 (1995), 129-145.  doi: 10.1007/BF00379918.  Google Scholar

[28]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 192 (2009), 187-215.  doi: 10.1007/s00205-008-0129-1.  Google Scholar

[29]

X. H. 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.  Google Scholar

[30]

X. H. 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.  Google Scholar

[31]

F. Poupaud, M. Rascle and J. P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differ. Equ., 123 (1995), 93-121.  doi: 10.1006/jdeq.1995.1158.  Google Scholar

[32]

A. Sitenko and V. Malnev, Plasma physics theory, Appl. Math. Math. Comput., 10, Chapman & Hall, Lonndon, 1995.  Google Scholar

[33]

Y. Sugiyama, Singularity formation for the 1-D compressible Euler equations with variable damping coefficient, Nonlinear Anal., 170 (2018), 70-87.  doi: 10.1016/j.na.2017.12.013.  Google Scholar

[34]

H. Sun, M. Mei and K. J. Zhang, Large time behaviors of solutions to the unipolar hydrodynamic model of semiconductors with physical boundary effect, Nonlinear Anal.-Real, 53 (2020), 103070. doi: 10.1016/j.nonrwa.2019.103070.  Google Scholar

[35]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Commun. Math. Phys., 157 (1993), 1-22.   Google Scholar

show all references

References:
[1]

K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron. Devices, 17 (1970), 38-47.   Google Scholar

[2]

S. G. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-up solutions to compressible Euler equations with time-dependent damping, J. Differ. Equ., 268 (2020), 5035-5077.  doi: 10.1016/j.jde.2019.11.002.  Google Scholar

[3]

G. Q. Chen and D. Wang, Convergence of shock schemes for the compressible Euler-Poisson equations, Commun. Math. Phys., 179 (1996), 333-364.   Google Scholar

[4]

P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29.  doi: 10.1016/0893-9659(90)90130-4.  Google Scholar

[5]

P. Degond and P. A. Markowich, A steady-state potential flow model for semiconductors, Ann. Math. Pure Appl., 4 (1993), 87-98.  doi: 10.1007/BF01765842.  Google Scholar

[6]

D. Donatelli, M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to Euler-Poisson equations for bipolar hydrodynamic model of semiconductors, J. Differ. Equ., 255 (2013), 3150-3184. doi: 10.1016/j.jde.2013.07.027.  Google Scholar

[7]

W. F. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differ. Equ., 133 (1997), 224-244.  doi: 10.1006/jdeq.1996.3203.  Google Scholar

[8]

I. M. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Commun. Partial Differ. Equ., 17 (1992), 553-577.  doi: 10.1080/03605309208820853.  Google Scholar

[9]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Ration. Mech. Anal., 179 (2006), 1-30.  doi: 10.1007/s00205-005-0369-2.  Google Scholar

[10]

I. Gasser, L. Hsiao and H. L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differ. Equ., 192 (2003), 326-359.  doi: 10.1016/S0022-0396(03)00122-0.  Google Scholar

[11]

L. Hsiao, P. A. Markowich and S. Wang, The asymptotic behavior of globally smooth solutions of the multidimensional isentropic hydrodynamic model for semiconductors, J. Differ. Equ., 192 (2003), 111-133.  doi: 10.1016/S0022-0396(03)00063-9.  Google Scholar

[12]

L. Hsiao and T. Yang, Asymptotic of Initial Boundary Value Problems for Hydrodynamic and Drift Diffusion Models for Semiconductors, J. Differ. Equ., 170 (2001), 473-493.  doi: 10.1006/jdeq.2000.3825.  Google Scholar

[13]

L. Hsiao and K. J. Zhang, The global weak solution and relaxation limits of the initialboundary value problem to the bipolar hydrodynamic model for semiconductors, Math. Models Meth. Appl. Sci., 10 (2000), 1333-1361.  doi: 10.1142/S0218202500000653.  Google Scholar

[14]

L. Hsiao and K. J. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations, J. Differ. Equ., 165 (2000), 315-354.  doi: 10.1006/jdeq.2000.3780.  Google Scholar

[15]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynamic model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429.  doi: 10.1137/100793025.  Google Scholar

[16]

F. M. Huang, M. 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.  Google Scholar

[17]

F. M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44 (2012), 1134-1164.  doi: 10.1137/110831647.  Google Scholar

[18]

F. M. Huang, M. Mei, Y. Wang and H. M. Yu, Asymptotic convergence to planar stationary waves for multi-dimensional unipolar hydrodynamic model of semiconductors, J. Differ. Equ., 251 (2011), 1305-1331.  doi: 10.1016/j.jde.2011.04.007.  Google Scholar

[19]

H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh Sect. A, 132 (2002), 359-378.  doi: 10.1017/S0308210500001670.  Google Scholar

[20]

H. L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic shock solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  doi: 10.1090/qam/1939010.  Google Scholar

[21]

H. T. Li, Large Time Behavior of Solutions to Hyperbolic Equations with Time-Dependent Damping (in Chinese), Ph.D thesis, Northeast Normal University, 2019. Google Scholar

[22]

H. T. Li, J. Y. Li, M. Mei and K. J. 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.  Google Scholar

[23]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830.  doi: 10.1137/S0036139996312168.  Google Scholar

[24]

L. P. Luan, M. Mei, B. Rubino and P. C. Zhu, Large-Time Behavior of Solutions to Cauchy Problem for Bipolar Euler-Poisson System with Time-Dependent Damping in Critical Case, preprint, 2020.  Google Scholar

[25]

P. A. Markowich, C. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-7091-6961-2.  Google Scholar

[26]

A. Matsumura, Global existence and asymptotics of the solutions of the second-order quasilinear hyperbolic equation with the first-order dissipation, Publ. RIMS Kyoto Univ., 13 (1977), 349-379.  doi: 10.2977/prims/1195189813.  Google Scholar

[27]

P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rat. Mech. Anal., 129 (1995), 129-145.  doi: 10.1007/BF00379918.  Google Scholar

[28]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a thermal hydrodynamic model for semiconductors, Arch. Ration. Mech. Anal., 192 (2009), 187-215.  doi: 10.1007/s00205-008-0129-1.  Google Scholar

[29]

X. H. 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.  Google Scholar

[30]

X. H. 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.  Google Scholar

[31]

F. Poupaud, M. Rascle and J. P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differ. Equ., 123 (1995), 93-121.  doi: 10.1006/jdeq.1995.1158.  Google Scholar

[32]

A. Sitenko and V. Malnev, Plasma physics theory, Appl. Math. Math. Comput., 10, Chapman & Hall, Lonndon, 1995.  Google Scholar

[33]

Y. Sugiyama, Singularity formation for the 1-D compressible Euler equations with variable damping coefficient, Nonlinear Anal., 170 (2018), 70-87.  doi: 10.1016/j.na.2017.12.013.  Google Scholar

[34]

H. Sun, M. Mei and K. J. Zhang, Large time behaviors of solutions to the unipolar hydrodynamic model of semiconductors with physical boundary effect, Nonlinear Anal.-Real, 53 (2020), 103070. doi: 10.1016/j.nonrwa.2019.103070.  Google Scholar

[35]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Commun. Math. Phys., 157 (1993), 1-22.   Google Scholar

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