American Institute of Mathematical Sciences

Advanced
March  2015, 8(1): 117-151. doi: 10.3934/krm.2015.8.117

Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate

Haifeng Hu 1, and  Kaijun Zhang 1,

1. 

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

Received  July 2014 Revised  August 2014 Published  December 2014

We study the Cauchy problem of a 1-D full hydrodynamic model for semiconductors where the energy equations are included. In the case of recombination-generation effects between electrons and holes being taken into consideration, the existence and uniqueness of a subsonic stationary solution of the related system are established. The convergence of the global smooth solution to the stationary solution exponentially is proved as time tends to infinity.
Keywords: Full hydrodynamic model, recombination-generation rate, Cauchy problem, smooth solution, asymptotic stability., energy estimates, stationary solution.
Mathematics Subject Classification: 35B40, 82D37, 35M3.
Citation: Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic and Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117
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), 572-587. doi: 10.1137/S0036141099355174.

[2]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[3]

S. Dimitrijev, Principles of Semiconductor Devices, Oxford University Press, 2nd edition, Oxford, 2011.

[4]

P. Degond and P. 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]

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), 3150-3184. doi: 10.1016/j.jde.2013.07.027.

[6]

W. Fang and K. Ito, Energy estimates for a one-dimensional hydrodynamic model of semiconductors, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00053-5.

[7]

W. Fang and K. Ito, Weak solutions to a one-dimensional hydrodynamic model of two carrier types for semiconductors, Nonlinear Anal., 28 (1997), 947-963. doi: 10.1016/0362-546X(95)00189-3.

[8]

I. Gasser, L. Hsiao and H. 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.

[9]

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

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[11]

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), 269-285. doi: 10.1007/s00605-002-0485-0.

[12]

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), 1595-1630. doi: 10.1137/100810228.

[13]

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), 411-429. doi: 10.1137/100793025.

[14]

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

[15]

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), 1134-1164. doi: 10.1137/110831647.

[16]

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), 777-796. doi: 10.1142/S0218202502001891.

[17]

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.

[18]

H. Hu and K. Zhang, Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors, Discrete Contin.Dyn.Syst.Ser.B, 19 (2014), 1601-1626. doi: 10.3934/dcdsb.2014.19.1601.

[19]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Besel-Boston-Berlin, 2001. doi: 10.1007/978-3-0348-8334-4.

[20]

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), 359-378. doi: 10.1017/S0308210500001670.

[21]

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), 810-830. doi: 10.1137/S0036139996312168.

[22]

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

[23]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[24]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[25]

N. J. Mauser, Y. Qiu and K. Zhang, Global existence and asymptotic limits of weak solutions of the bipolar hydrodynamic model for semiconductors, Monatsh. Math., 140 (2003), 285-313. doi: 10.1007/s00605-002-0543-7.

[26]

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.

[27]

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.

[28]

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

[29]

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.

[30]

Y. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045.

[31]

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

[32]

D. Wang, Global solutions to the Euler-Poisson equations of two-carrier types in one dimension, Z. Angew. Math. Phys., 48 (1997), 680-693. doi: 10.1007/s000330050056.

[33]

K. Zhang, On the initial-boundary value problem for the bipolar hydrodynamic model for semiconductors, J. Differential Equations, 171 (2001), 251-293. doi: 10.1006/jdeq.2000.3850.

[34]

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.

[35]

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.

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), 572-587. doi: 10.1137/S0036141099355174.

[2]

G. Alì and A. Jüngel, Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas, J. Differential Equations, 190 (2003), 663-685. doi: 10.1016/S0022-0396(02)00157-2.

[3]

S. Dimitrijev, Principles of Semiconductor Devices, Oxford University Press, 2nd edition, Oxford, 2011.

[4]

P. Degond and P. 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]

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), 3150-3184. doi: 10.1016/j.jde.2013.07.027.

[6]

W. Fang and K. Ito, Energy estimates for a one-dimensional hydrodynamic model of semiconductors, Appl. Math. Lett., 9 (1996), 65-70. doi: 10.1016/0893-9659(96)00053-5.

[7]

W. Fang and K. Ito, Weak solutions to a one-dimensional hydrodynamic model of two carrier types for semiconductors, Nonlinear Anal., 28 (1997), 947-963. doi: 10.1016/0362-546X(95)00189-3.

[8]

I. Gasser, L. Hsiao and H. 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.

[9]

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

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981.

[11]

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), 269-285. doi: 10.1007/s00605-002-0485-0.

[12]

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), 1595-1630. doi: 10.1137/100810228.

[13]

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), 411-429. doi: 10.1137/100793025.

[14]

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

[15]

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), 1134-1164. doi: 10.1137/110831647.

[16]

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), 777-796. doi: 10.1142/S0218202502001891.

[17]

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.

[18]

H. Hu and K. Zhang, Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors, Discrete Contin.Dyn.Syst.Ser.B, 19 (2014), 1601-1626. doi: 10.3934/dcdsb.2014.19.1601.

[19]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Verlag, Besel-Boston-Berlin, 2001. doi: 10.1007/978-3-0348-8334-4.

[20]

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), 359-378. doi: 10.1017/S0308210500001670.

[21]

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), 810-830. doi: 10.1137/S0036139996312168.

[22]

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

[23]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. doi: 10.1007/978-1-4612-1116-7.

[24]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[25]

N. J. Mauser, Y. Qiu and K. Zhang, Global existence and asymptotic limits of weak solutions of the bipolar hydrodynamic model for semiconductors, Monatsh. Math., 140 (2003), 285-313. doi: 10.1007/s00605-002-0543-7.

[26]

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.

[27]

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.

[28]

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

[29]

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.

[30]

Y. Peng and J. Xu, Global well-posedness of the hydrodynamic model for two-carrier plasmas, J. Differential Equations, 255 (2013), 3447-3471. doi: 10.1016/j.jde.2013.07.045.

[31]

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

[32]

D. Wang, Global solutions to the Euler-Poisson equations of two-carrier types in one dimension, Z. Angew. Math. Phys., 48 (1997), 680-693. doi: 10.1007/s000330050056.

[33]

K. Zhang, On the initial-boundary value problem for the bipolar hydrodynamic model for semiconductors, J. Differential Equations, 171 (2001), 251-293. doi: 10.1006/jdeq.2000.3850.

[34]

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.

[35]

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.

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