American Institute of Mathematical Sciences

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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 & 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. 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. doi: 10.1016/S0022-0396(02)00157-2. Google Scholar

[3]

S. Dimitrijev, Principles of Semiconductor Devices,, Oxford University Press, (2011). Google Scholar

[4]

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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-005-0369-2. Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.3934/dcdsb.2014.19.1601. Google Scholar

[19]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (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. 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. doi: 10.1137/S0036139996312168. Google Scholar

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0022-0396(02)00157-2. Google Scholar

[3]

S. Dimitrijev, Principles of Semiconductor Devices,, Oxford University Press, (2011). Google Scholar

[4]

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. 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. 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. 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. 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. 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. doi: 10.1007/s00205-005-0369-2. Google Scholar

[10]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (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. 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. 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. 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. 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. 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. 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. 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. doi: 10.3934/dcdsb.2014.19.1601. Google Scholar

[19]

A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (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. 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. doi: 10.1137/S0036139996312168. Google Scholar

[22]

P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (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, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1006/jdeq.2000.3799. Google Scholar

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