# American Institute of Mathematical Sciences

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. doi: 10.1137/S0036141099355174. [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. [3] S. Dimitrijev, Principles of Semiconductor Devices,, Oxford University Press, (2011). [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. [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. [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. [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. [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. [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. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). [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. [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. [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. [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. [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. [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. [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. [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. [19] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4. [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. [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. [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. [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. [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. [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. [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. [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. [28] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639. [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. [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. [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. [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. [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. [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. [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.

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. [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. [3] S. Dimitrijev, Principles of Semiconductor Devices,, Oxford University Press, (2011). [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. [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. [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. [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. [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. [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. [10] D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981). [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. [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. [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. [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. [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. [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. [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. [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. [19] A. Jüngel, Quasi-Hydrodynamic Semiconductor Equations,, Progress in Nonlinear Differential Equations and their Applications, (2001). doi: 10.1007/978-3-0348-8334-4. [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. [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. [22] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990). doi: 10.1007/978-3-7091-6961-2. [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. [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. [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. [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. [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. [28] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors,, Osaka J. Math., 44 (2007), 639. [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. [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. [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. [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. [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. [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. [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.
 [1] Shaoyong Lai, Yong Hong Wu. The asymptotic solution of the Cauchy problem for a generalized Boussinesq equation. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 401-408. doi: 10.3934/dcdsb.2003.3.401 [2] Belkacem Said-Houari, Radouane Rahali. Asymptotic behavior of the solution to the Cauchy problem for the Timoshenko system in thermoelasticity of type III. Evolution Equations & Control Theory, 2013, 2 (2) : 423-440. doi: 10.3934/eect.2013.2.423 [3] Boling Guo, Guangwu Wang. Existence of the solution for the viscous bipolar quantum hydrodynamic model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3183-3210. doi: 10.3934/dcds.2017136 [4] Chengxia Lei, Yihong Du. Asymptotic profile of the solution to a free boundary problem arising in a shifting climate model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 895-911. doi: 10.3934/dcdsb.2017045 [5] Ellen Baake, Michael Baake, Majid Salamat. The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 63-95. doi: 10.3934/dcds.2016.36.63 [6] Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991 [7] Jiang Xu. Well-posedness and stability of classical solutions to the multidimensional full hydrodynamic model for semiconductors. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1073-1092. doi: 10.3934/cpaa.2009.8.1073 [8] 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 [9] Ghendrih Philippe, Hauray Maxime, Anne Nouri. Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solution. Kinetic & Related Models, 2009, 2 (4) : 707-725. doi: 10.3934/krm.2009.2.707 [10] Yongming Liu, Lei Yao. Global solution and decay rate for a reduced gravity two and a half layer model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2613-2638. doi: 10.3934/dcdsb.2018267 [11] Ellen Baake, Michael Baake, Majid Salamat. Erratum and addendum to: The general recombination equation in continuous time and its solution. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2365-2366. doi: 10.3934/dcds.2016.36.2365 [12] Minhajul, T. Raja Sekhar, G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3389-3408. doi: 10.3934/cpaa.2019153 [13] Ling Mi. Asymptotic behavior for the unique positive solution to a singular elliptic problem. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1053-1072. doi: 10.3934/cpaa.2015.14.1053 [14] Masataka Shibata. Asymptotic shape of a solution for the Plasma problem in higher dimensional spaces. Communications on Pure & Applied Analysis, 2003, 2 (2) : 259-275. doi: 10.3934/cpaa.2003.2.259 [15] Qiao Liu, Shangbin Cui. Regularizing rate estimates for mild solutions of the incompressible Magneto-hydrodynamic system. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1643-1660. doi: 10.3934/cpaa.2012.11.1643 [16] Francesca Crispo, Paolo Maremonti. A remark on the partial regularity of a suitable weak solution to the Navier-Stokes Cauchy problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1283-1294. doi: 10.3934/dcds.2017053 [17] Libin Wang. Breakdown of $C^1$ solution to the Cauchy problem for quasilinear hyperbolic systems with characteristics with constant multiplicity. Communications on Pure & Applied Analysis, 2003, 2 (1) : 77-89. doi: 10.3934/cpaa.2003.2.77 [18] Roman Chapko, B. Tomas Johansson. On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach. Inverse Problems & Imaging, 2012, 6 (1) : 25-38. doi: 10.3934/ipi.2012.6.25 [19] Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems & Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427 [20] Shijin Ding, Changyou Wang, Huanyao Wen. Weak solution to compressible hydrodynamic flow of liquid crystals in dimension one. Discrete & Continuous Dynamical Systems - B, 2011, 15 (2) : 357-371. doi: 10.3934/dcdsb.2011.15.357

2017 Impact Factor: 1.219