Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors

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August 2014, 19(6): 1601-1626. doi: 10.3934/dcdsb.2014.19.1601

Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors

Haifeng Hu ^1, and Kaijun Zhang ^1,

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

Received December 2013 Revised January 2014 Published June 2014

Abstract
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We consider the initial boundary value problem of the one dimensional full bipolar hydrodynamic model for semiconductors. The existence and uniqueness of the stationary solution are established by the theory of strongly elliptic systems and the Banach fixed point theorem. The exponentially asymptotic stability of the stationary solution is given by means of the energy estimate method.

Keywords: asymptotic stability, Full hydrodynamic model, stationary solution, strongly elliptic systems., energy estimate.

Mathematics Subject Classification: 35B40, 35M1.

Citation: Haifeng Hu, Kaijun Zhang. Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1601-1626. doi: 10.3934/dcdsb.2014.19.1601

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]	K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), 38-47. doi: 10.1109/T-ED.1970.16921.
[3]	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.
[4]	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.
[5]	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.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.
[7]	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.
[8]	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.
[9]	F. Huang and Y. 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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	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.
[15]	S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, in Analysis of systems of conservation laws, Chapman & Hall/CRC, Monogr. Surv. Pure Appl. Math., 99, (1999), 87-127.
[16]	S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Rational Mech. Anal., 170 (2003), 297-329. doi: 10.1007/s00205-003-0273-6.
[17]	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.
[18]	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.
[19]	Y. Li, Global existence and asymptotic behavior of smooth solutions to a bipolar Euler-Poisson equation in a bound domain, Z. Angew. Math. Phys., 64 (2013), 1125-1144. doi: 10.1007/s00033-012-0269-x.
[20]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[21]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.
[22]	M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain, Kinetic and Related Models, 5 (2012), 537-550. doi: 10.3934/krm.2012.5.537.
[23]	R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.
[24]	S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665.
[25]	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.
[26]	S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Commun. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.
[27]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[28]	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.
[29]	J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511662850.
[30]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
[31]	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.

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]	K. Bløtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), 38-47. doi: 10.1109/T-ED.1970.16921.
[3]	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.
[4]	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.
[5]	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.
[6]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.
[7]	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.
[8]	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.
[9]	F. Huang and Y. 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.
[10]	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.
[11]	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.
[12]	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.
[13]	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.
[14]	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.
[15]	S. Kawashima, Y. Nikkuni and S. Nishibata, The initial value problem for hyperbolic-elliptic coupled systems and applications to radiation hydrodynamics, in Analysis of systems of conservation laws, Chapman & Hall/CRC, Monogr. Surv. Pure Appl. Math., 99, (1999), 87-127.
[16]	S. Kawashima, Y. Nikkuni and S. Nishibata, Large-time behavior of solutions to hyperbolic-elliptic coupled systems, Arch. Rational Mech. Anal., 170 (2003), 297-329. doi: 10.1007/s00205-003-0273-6.
[17]	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.
[18]	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.
[19]	Y. Li, Global existence and asymptotic behavior of smooth solutions to a bipolar Euler-Poisson equation in a bound domain, Z. Angew. Math. Phys., 64 (2013), 1125-1144. doi: 10.1007/s00033-012-0269-x.
[20]	W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[21]	P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations, Springer-Verlag, Vienna, 1990. doi: 10.1007/978-3-7091-6961-2.
[22]	M. Mei, B. Rubino and R. Sampalmieri, Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain, Kinetic and Related Models, 5 (2012), 537-550. doi: 10.3934/krm.2012.5.537.
[23]	R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.
[24]	S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665.
[25]	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.
[26]	S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit, Commun. Math. Phys., 104 (1986), 49-75. doi: 10.1007/BF01210792.
[27]	R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, $2^{nd}$ edition, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3.
[28]	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.
[29]	J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511662850.
[30]	E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
[31]	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.

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