# American Institute of Mathematical Sciences

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

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

Received  December 2013 Revised  January 2014 Published  June 2014

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.
Citation: Haifeng Hu, Kaijun Zhang. Analysis on the initial-boundary value problem of a full bipolar hydrodynamic model for semiconductors. Discrete & 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.  doi: 10.1137/S0036141099355174.  Google Scholar [2] K. Bløtekjær, Transport equations for electrons in two-valley semiconductors,, IEEE Trans. Electron Devices, 17 (1970), 38.  doi: 10.1109/T-ED.1970.16921.  Google Scholar [3] 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 [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.  doi: 10.1016/j.jde.2013.07.027.  Google Scholar [5] 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 [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar [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.  doi: 10.1007/s00605-002-0485-0.  Google Scholar [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.  doi: 10.1142/S0218202502001891.  Google Scholar [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.  doi: 10.3934/dcds.2009.24.455.  Google Scholar [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.  doi: 10.1137/100810228.  Google Scholar [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.  doi: 10.1137/110831647.  Google Scholar [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.  doi: 10.1137/100793025.  Google Scholar [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.  doi: 10.1016/j.jde.2011.04.007.  Google Scholar [14] 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 [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, 99 (1999), 87.   Google Scholar [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.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [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.  doi: 10.1137/S0036139996312168.  Google Scholar [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.  doi: 10.1017/S0308210500001670.  Google Scholar [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.  doi: 10.1007/s00033-012-0269-x.  Google Scholar [20] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).   Google Scholar [21] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar [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.  doi: 10.3934/krm.2012.5.537.  Google Scholar [23] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation,, J. Math. Anal. Appl., 198 (1996), 262.  doi: 10.1006/jmaa.1996.0081.  Google Scholar [24] 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 [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.  doi: 10.1007/s00205-008-0129-1.  Google Scholar [26] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Commun. Math. Phys., 104 (1986), 49.  doi: 10.1007/BF01210792.  Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [28] 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 [29] J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511662850.  Google Scholar [30] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4612-0985-0.  Google Scholar [31] 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

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] K. Bløtekjær, Transport equations for electrons in two-valley semiconductors,, IEEE Trans. Electron Devices, 17 (1970), 38.  doi: 10.1109/T-ED.1970.16921.  Google Scholar [3] 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 [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.  doi: 10.1016/j.jde.2013.07.027.  Google Scholar [5] 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 [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, Reprint of the 1998 edition, (1998).   Google Scholar [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.  doi: 10.1007/s00605-002-0485-0.  Google Scholar [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.  doi: 10.1142/S0218202502001891.  Google Scholar [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.  doi: 10.3934/dcds.2009.24.455.  Google Scholar [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.  doi: 10.1137/100810228.  Google Scholar [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.  doi: 10.1137/110831647.  Google Scholar [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.  doi: 10.1137/100793025.  Google Scholar [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.  doi: 10.1016/j.jde.2011.04.007.  Google Scholar [14] 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 [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, 99 (1999), 87.   Google Scholar [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.  doi: 10.1007/s00205-003-0273-6.  Google Scholar [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.  doi: 10.1137/S0036139996312168.  Google Scholar [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.  doi: 10.1017/S0308210500001670.  Google Scholar [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.  doi: 10.1007/s00033-012-0269-x.  Google Scholar [20] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations,, Cambridge University Press, (2000).   Google Scholar [21] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).  doi: 10.1007/978-3-7091-6961-2.  Google Scholar [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.  doi: 10.3934/krm.2012.5.537.  Google Scholar [23] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation,, J. Math. Anal. Appl., 198 (1996), 262.  doi: 10.1006/jmaa.1996.0081.  Google Scholar [24] 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 [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.  doi: 10.1007/s00205-008-0129-1.  Google Scholar [26] S. Schochet, The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit,, Commun. Math. Phys., 104 (1986), 49.  doi: 10.1007/BF01210792.  Google Scholar [27] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics,, $2^{nd}$ edition, (1997).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar [28] 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 [29] J. T. Wloka, B. Rowley and B. Lawruk, Boundary Value Problems for Elliptic Systems,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511662850.  Google Scholar [30] E. Zeidler, Nonlinear Functional Analysis and Its Applications. II/A. Linear Monotone Operators,, Springer-Verlag, (1990).  doi: 10.1007/978-1-4612-0985-0.  Google Scholar [31] 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
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