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

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.
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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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