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

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  March 2021 Early access  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.

Citation: Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure and 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. 

[2]

S. G. ChenH. LiJ. LiM. 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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

I. GasserL. 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.

[11]

L. HsiaoP. 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.

[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.

[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.

[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.

[15]

F. M. HuangM. MeiY. 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.

[16]

F. M. 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.

[17]

F. M. HuangM. MeiY. 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.

[18]

F. M. HuangM. MeiY. 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.

[19]

H. L. LiP. 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.

[20]

H. L. LiP. 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.

[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.

[22]

H. T. LiJ. Y. LiM. 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.

[23]

T. LuoR. 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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

F. PoupaudM. 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.

[32]

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

[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.

[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.

[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. 

show all references

References:
[1]

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

[2]

S. G. ChenH. LiJ. LiM. 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.

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[10]

I. GasserL. 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.

[11]

L. HsiaoP. 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.

[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.

[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.

[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.

[15]

F. M. HuangM. MeiY. 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.

[16]

F. M. 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.

[17]

F. M. HuangM. MeiY. 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.

[18]

F. M. HuangM. MeiY. 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.

[19]

H. L. LiP. 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.

[20]

H. L. LiP. 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.

[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.

[22]

H. T. LiJ. Y. LiM. 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.

[23]

T. LuoR. 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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[31]

F. PoupaudM. 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.

[32]

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

[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.

[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.

[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. 

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