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

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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

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

Received July 2014 Revised August 2014 Published December 2014

Abstract
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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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