August  2014, 19(6): 1601-1626. doi: 10.3934/dcdsb.2014.19.1601

Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun, MO 130024, China, China

Received  December 2013 Revised  January 2014 Published  June 2014

We consider the initial boundary value problem of the one dimensional full bipolar hydrodynamic model for semiconductors. The existence and uniqueness of the stationary solution are established by the theory of strongly elliptic systems and the Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is given by means of the energy estimate method.
Citation: Haifeng Hu, Kaijun Zhang. Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1601-1626. doi: 10.3934/dcdsb.2014.19.1601
References:
[1]

G. Ali, D. Bini and S. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors,, SIAM J. Math. Anal., 32 (2000), 572. doi: 10.1137/S0036141099355174.

[2]

K. Bløtekjær, Transport equations for electrons in two-valley semiconductors,, IEEE Trans. Electron Devices, 17 (1970), 38. doi: 10.1109/T-ED.1970.16921.

[3]

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

[4]

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

[5]

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

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).

[7]

L. Hsiao, S. Jiang and P. Zhang, Global existence and exponential stability of smooth solutions to a full hydrodynamic model to semiconductors,, Monatsh. Math., 136 (2002), 269. doi: 10.1007/s00605-002-0485-0.

[8]

L. Hsiao and S. Wang, Asymptotic behavior of global smooth solutions to the full 1D hydrodynamic model for semiconductors,, Math.Models Methods Appl.Sci., 12 (2002), 777. doi: 10.1142/S0218202502001891.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4.

[15]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in Analysis of systems of conservation laws, 99 (1999), 87.

[16]

S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Rational Mech. Anal., 170 (2003), 297. doi: 10.1007/s00205-003-0273-6.

[17]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors,, SIAM J. Appl. Math., 59 (1999), 810. doi: 10.1137/S0036139996312168.

[18]

H. Li, P. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359. doi: 10.1017/S0308210500001670.

[19]

Y. Li, Global existence and asymptotic behavior of smooth solutions to a bipolar Euler-Poisson equation in a bound domain,, Z. Angew. Math. Phys., 64 (2013), 1125. doi: 10.1007/s00033-012-0269-x.

[20]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).

[21]

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

[22]

M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain,, Kinetic and Related Models, 5 (2012), 537. doi: 10.3934/krm.2012.5.537.

[23]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation,, J. Math. Anal. Appl., 198 (1996), 262. doi: 10.1006/jmaa.1996.0081.

[24]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639.

[25]

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

[26]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Commun. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[28]

N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors,, Nonlinear Anal., 73 (2010), 779. doi: 10.1016/j.na.2010.04.015.

[29]

J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511662850.

[30]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.

[31]

K. Zhang, On the initial-boundary value problem for the bipolar hydrodynamic model for semiconductors,, J. Differential Equations, 171 (2001), 251. doi: 10.1006/jdeq.2000.3850.

show all references

References:
[1]

G. Ali, D. Bini and S. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Poisson model for semiconductors,, SIAM J. Math. Anal., 32 (2000), 572. doi: 10.1137/S0036141099355174.

[2]

K. Bløtekjær, Transport equations for electrons in two-valley semiconductors,, IEEE Trans. Electron Devices, 17 (1970), 38. doi: 10.1109/T-ED.1970.16921.

[3]

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

[4]

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

[5]

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

[6]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).

[7]

L. Hsiao, S. Jiang and P. Zhang, Global existence and exponential stability of smooth solutions to a full hydrodynamic model to semiconductors,, Monatsh. Math., 136 (2002), 269. doi: 10.1007/s00605-002-0485-0.

[8]

L. Hsiao and S. Wang, Asymptotic behavior of global smooth solutions to the full 1D hydrodynamic model for semiconductors,, Math.Models Methods Appl.Sci., 12 (2002), 777. doi: 10.1142/S0218202502001891.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4.

[15]

S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics,, in Analysis of systems of conservation laws, 99 (1999), 87.

[16]

S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems,, Arch. Rational Mech. Anal., 170 (2003), 297. doi: 10.1007/s00205-003-0273-6.

[17]

T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors,, SIAM J. Appl. Math., 59 (1999), 810. doi: 10.1137/S0036139996312168.

[18]

H. Li, P. Markowich and M. Mei, Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors,, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 359. doi: 10.1017/S0308210500001670.

[19]

Y. Li, Global existence and asymptotic behavior of smooth solutions to a bipolar Euler-Poisson equation in a bound domain,, Z. Angew. Math. Phys., 64 (2013), 1125. doi: 10.1007/s00033-012-0269-x.

[20]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).

[21]

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

[22]

M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain,, Kinetic and Related Models, 5 (2012), 537. doi: 10.3934/krm.2012.5.537.

[23]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation,, J. Math. Anal. Appl., 198 (1996), 262. doi: 10.1006/jmaa.1996.0081.

[24]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639.

[25]

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

[26]

S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Commun. Math. Phys., 104 (1986), 49. doi: 10.1007/BF01210792.

[27]

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997). doi: 10.1007/978-1-4612-0645-3.

[28]

N. Tsuge, Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors,, Nonlinear Anal., 73 (2010), 779. doi: 10.1016/j.na.2010.04.015.

[29]

J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511662850.

[30]

E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators,, Springer-Verlag, (1990). doi: 10.1007/978-1-4612-0985-0.

[31]

K. Zhang, On the initial-boundary value problem for the bipolar hydrodynamic model for semiconductors,, J. Differential Equations, 171 (2001), 251. doi: 10.1006/jdeq.2000.3850.

[1]

Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic & Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117

[2]

Jiang Xu. Well-posedness and stability of classical solutions to the multidimensional full hydrodynamic model for semiconductors. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1073-1092. doi: 10.3934/cpaa.2009.8.1073

[3]

Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136

[4]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630

[5]

Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic & Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537

[6]

Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146

[7]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i

[8]

Ming Mei, Yong Wang. Stability of stationary waves for full Euler-Poisson system in multi-dimensional space. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1775-1807. doi: 10.3934/cpaa.2012.11.1775

[9]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991

[10]

Christopher E. Elmer. The stability of stationary fronts for a discrete nerve axon model. Mathematical Biosciences & Engineering, 2007, 4 (1) : 113-129. doi: 10.3934/mbe.2007.4.113

[11]

Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707

[12]

Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053

[13]

Francesca R. Guarguaglini. Stationary solutions and asymptotic behaviour for a chemotaxis hyperbolic model on a network. Networks & Heterogeneous Media, 2018, 13 (1) : 47-67. doi: 10.3934/nhm.2018003

[14]

Ken Shirakawa. Asymptotic stability for dynamical systems associated with the one-dimensional Frémond model of shape memory alloys. Conference Publications, 2003, 2003 (Special) : 798-808. doi: 10.3934/proc.2003.2003.798

[15]

Jie Jiang, Boling Guo. Asymptotic behavior of solutions to a one-dimensional full model for phase transitions with microscopic movements. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 167-190. doi: 10.3934/dcds.2012.32.167

[16]

Philippe Jouan, Said Naciri. Asymptotic stability of uniformly bounded nonlinear switched systems. Mathematical Control & Related Fields, 2013, 3 (3) : 323-345. doi: 10.3934/mcrf.2013.3.323

[17]

Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245

[18]

Xianjin Chen, Jianxin Zhou. A local min-orthogonal method for multiple solutions of strongly coupled elliptic systems. Conference Publications, 2009, 2009 (Special) : 151-160. doi: 10.3934/proc.2009.2009.151

[19]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1

[20]

Masahiro Suzuki. Asymptotic stability of stationary solutions to the Euler-Poisson equations arising in plasma physics. Kinetic & Related Models, 2011, 4 (2) : 569-588. doi: 10.3934/krm.2011.4.569

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]