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  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 & 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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  Google Scholar

[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.  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. 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.  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

[1]

Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021062

[2]

Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195

[3]

Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222

[4]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007

[5]

Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044

[6]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[7]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[8]

Michiyuki Watanabe. Inverse $N$-body scattering with the time-dependent hartree-fock approximation. Inverse Problems & Imaging, 2021, 15 (3) : 499-517. doi: 10.3934/ipi.2021002

[9]

Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249

[10]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002

[11]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[12]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103

[13]

Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034

[14]

Guillermo Reyes, Juan-Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with slowly decaying density. Communications on Pure & Applied Analysis, 2009, 8 (2) : 493-508. doi: 10.3934/cpaa.2009.8.493

[15]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

[16]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[17]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183

[18]

Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256

[19]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems & Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[20]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (99)
  • HTML views (149)
  • Cited by (0)

Other articles
by authors

[Back to Top]