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September  2012, 5(3): 537-550. doi: 10.3934/krm.2012.5.537

Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain

1. 

Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada

2. 

Department of Pure and Applied Mathematics,University of L'Aquila, 67010 Coppito, L'Aquila, Italy, Italy

Received  March 2012 Revised  May 2012 Published  August 2012

In this paper we present a physically relevant hydrodynamic model for a bipolar semiconductor device considering Ohmic conductor boundary conditions and a non-flat doping profile. For such an Euler-Poisson system, we prove, by means of a technical energy method, that the solutions are unique, exist globally and asymptotically converge to the corresponding stationary solutions. An exponential decay rate is also derived. Moreover we allow that the two pressure functions can be different.
Citation: Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic and Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537
References:
[1]

G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[3]

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

[4]

G. Chen, J. Jerome and B. Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, Nonlinear Anal., 30 (1997), 233-244. doi: 10.1016/S0362-546X(96)00198-8.

[5]

G. Chen and D. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.

[6]

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.

[7]

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

[8]

I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.

[9]

I. Gasser, L. Hsiao and H.-L. 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.

[10]

I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1996), 269-282. doi: 10.1.1.53.9991.

[11]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.

[12]

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

[13]

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

[14]

F.-M. Huang and Y.-P. 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), 455-470. doi: 10.3934/dcds.2009.24.455.

[15]

F.-M. 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), 1595-1630. doi: 10.1137/100810228.

[16]

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.

[17]

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

[18]

F.-M. 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), 1305-1331. doi: 10.1016/j.jde.2011.04.007.

[19]

J. W. Jerome, Steady Euler-Poisson system: a differential/integral equation formulation with general constitutive relations, Nonlinear Anal., 71 (2009), e2188-e2193. doi: 10.1016/j.na.2009.04.042.

[20]

A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.

[21]

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.

[22]

H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.

[23]

C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.

[24]

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: 10.1.1.55.4600.

[25]

P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2005), suppl. 2, S224-S240 doi: 10.1007/s00021-005-0155-9.

[26]

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

[27]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.

[28]

M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.

[29]

M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.

[30]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.

[31]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.

[32]

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

[33]

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

[34]

A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman & Hall, London, 1995.

[35]

N. Tsuge, Existence and uniqueness of stationary solutions to one-dimensional bipolar hydrodynamic model of semiconductors, Nonlinear Analysis, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015.

[36]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.

[37]

C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799.

[38]

C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.

show all references

References:
[1]

G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.

[2]

G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.

[3]

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

[4]

G. Chen, J. Jerome and B. Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, Nonlinear Anal., 30 (1997), 233-244. doi: 10.1016/S0362-546X(96)00198-8.

[5]

G. Chen and D. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.

[6]

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.

[7]

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

[8]

I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.

[9]

I. Gasser, L. Hsiao and H.-L. 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.

[10]

I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1996), 269-282. doi: 10.1.1.53.9991.

[11]

Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.

[12]

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

[13]

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

[14]

F.-M. Huang and Y.-P. 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), 455-470. doi: 10.3934/dcds.2009.24.455.

[15]

F.-M. 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), 1595-1630. doi: 10.1137/100810228.

[16]

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.

[17]

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

[18]

F.-M. 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), 1305-1331. doi: 10.1016/j.jde.2011.04.007.

[19]

J. W. Jerome, Steady Euler-Poisson system: a differential/integral equation formulation with general constitutive relations, Nonlinear Anal., 71 (2009), e2188-e2193. doi: 10.1016/j.na.2009.04.042.

[20]

A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.

[21]

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.

[22]

H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.

[23]

C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.

[24]

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: 10.1.1.55.4600.

[25]

P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2005), suppl. 2, S224-S240 doi: 10.1007/s00021-005-0155-9.

[26]

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

[27]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.

[28]

M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.

[29]

M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.

[30]

R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.

[31]

S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.

[32]

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

[33]

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

[34]

A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman & Hall, London, 1995.

[35]

N. Tsuge, Existence and uniqueness of stationary solutions to one-dimensional bipolar hydrodynamic model of semiconductors, Nonlinear Analysis, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015.

[36]

B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.

[37]

C. Zhu and H. Hattori, Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species, J. Differential Equations, 166 (2000), 1-32. doi: 10.1006/jdeq.2000.3799.

[38]

C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.

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