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September  2012, 5(3): 537-550. doi: 10.3934/krm.2012.5.537

## Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain

 1 Department of Mathematics, Champlain College Saint-Lambert, Quebec, J4P 3P2, Canada 2 Department of Pure and Applied Mathematics,University of L'Aquila, 67010 Coppito, L'Aquila, Italy, Italy

Received  March 2012 Revised  May 2012 Published  August 2012

In this paper we present a physically relevant hydrodynamic model for a bipolar semiconductor device considering Ohmic conductor boundary conditions and a non-flat doping profile. For such an Euler-Poisson system, we prove, by means of a technical energy method, that the solutions are unique, exist globally and asymptotically converge to the corresponding stationary solutions. An exponential decay rate is also derived. Moreover we allow that the two pressure functions can be different.
Citation: Ming Mei, Bruno Rubino, Rosella Sampalmieri. Asymptotic behavior of solutions to the bipolar hydrodynamic model of semiconductors in bounded domain. Kinetic & Related Models, 2012, 5 (3) : 537-550. doi: 10.3934/krm.2012.5.537
##### References:
 [1] G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.  Google Scholar [2] G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.  Google Scholar [3] K. Blφtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), 38-47. Google Scholar [4] G. Chen, J. Jerome and B. Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, Nonlinear Anal., 30 (1997), 233-244. doi: 10.1016/S0362-546X(96)00198-8.  Google Scholar [5] G. Chen and D. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.  Google Scholar [6] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.  Google Scholar [7] W. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differential Equations, 133 (1997), 224-244. doi: 10.1006/jdeq.1996.3203.  Google Scholar [8] I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.  Google Scholar [9] I. Gasser, L. Hsiao and H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359. doi: 10.1016/S0022-0396(03)00122-0.  Google Scholar [10] I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1996), 269-282. doi: 10.1.1.53.9991.  Google Scholar [11] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar [12] 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), 1333-1361. doi: 10.1142/S0218202500000653.  Google Scholar [13] L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar [14] F.-M. Huang and Y.-P. 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.  Google Scholar [15] F.-M. 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.  Google Scholar [16] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44, (2012), 1134-1164. doi: 10.1137/110831647.  Google Scholar [17] F.-M. Huang, M. Mei, Y. Wang and H. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynami model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.  Google Scholar [18] F.-M. 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.  Google Scholar [19] J. W. Jerome, Steady Euler-Poisson system: a differential/integral equation formulation with general constitutive relations, Nonlinear Anal., 71 (2009), e2188-e2193. doi: 10.1016/j.na.2009.04.042.  Google Scholar [20] A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.  Google Scholar [21] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh, Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar [22] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  Google Scholar [23] C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.  Google Scholar [24] T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1.1.55.4600.  Google Scholar [25] P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2005), suppl. 2, S224-S240 doi: 10.1007/s00021-005-0155-9.  Google Scholar [26] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.  Google Scholar [27] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.  Google Scholar [28] M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.  Google Scholar [29] M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.  Google Scholar [30] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.  Google Scholar [31] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.  Google Scholar [32] 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.  Google Scholar [33] F. Poupaud, M. Rascle and J.-P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential Equations, 123 (1995), 93-121. doi: 10.1006/jdeq.1995.1158.  Google Scholar [34] A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman $&$ Hall, London, 1995.  Google Scholar [35] N. Tsuge, Existence and uniqueness of stationary solutions to one-dimensional bipolar hydrodynamic model of semiconductors, Nonlinear Analysis, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015.  Google Scholar [36] B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.  Google Scholar [37] 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), 1-32. doi: 10.1006/jdeq.2000.3799.  Google Scholar [38] C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.  Google Scholar

show all references

##### References:
 [1] G. Ali, Global existence of smooth solutions of the N-dimensional Euler-Possion model, SIAM J. Math. Anal., 35 (2003), 389-422. doi: 10.1137/S0036141001393225.  Google Scholar [2] G. Ali, D. Bini and D. Rionero, Global existence and relaxation limit for smooth solutions to the Euler-Possion model for semiconductors, SIAM J. Math. Anal., 32 (2000), 572-587. doi: 10.1137/S0036141099355174.  Google Scholar [3] K. Blφtekjær, Transport equations for electrons in two-valley semiconductors, IEEE Trans. Electron Devices, 17 (1970), 38-47. Google Scholar [4] G. Chen, J. Jerome and B. Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, Nonlinear Anal., 30 (1997), 233-244. doi: 10.1016/S0362-546X(96)00198-8.  Google Scholar [5] G. Chen and D. Wang, Convergence of shock capturing schemes for the compressible Euler-Poisson equations, Comm. Math. Phys., 179 (1996), 333-364. doi: 10.1007/BF02102592.  Google Scholar [6] P. Degond and P. A. Markowich, On a one-dimensional steady-state hydrodynamic model, Appl. Math. Lett., 3 (1990), 25-29. doi: 10.1016/0893-9659(90)90130-4.  Google Scholar [7] W. Fang and K. Ito, Steady-state solutions of a one-dimensional hydrodynamic model for semiconductors, J. Differential Equations, 133 (1997), 224-244. doi: 10.1006/jdeq.1996.3203.  Google Scholar [8] I. Gamba, Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductor, Comm. Partial Diff. Eqns, 17 (1992), 553-577. doi: 10.1080/03605309208820853.  Google Scholar [9] I. Gasser, L. Hsiao and H.-L. Li, Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors, J. Differential Equations, 192 (2003), 326-359. doi: 10.1016/S0022-0396(03)00122-0.  Google Scholar [10] I. Gasser and R. Natalini, The energy transport and the drift diffusion equations as relaxation limits of the hydrodynamic model for semiconductors, Quart. Appl. Math., 57 (1996), 269-282. doi: 10.1.1.53.9991.  Google Scholar [11] Y. Guo and W. Strauss, Stability of semiconductor states with insulating and contact boundary conditions, Arch. Rational Mech. Anal., 179 (2005), 1-30. doi: 10.1007/s00205-005-0369-2.  Google Scholar [12] 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), 1333-1361. doi: 10.1142/S0218202500000653.  Google Scholar [13] L. Hsiao and K. Zhang, The relaxation of the hydrodynamic model for semiconductors to drift diffusion equations, J. Differential Equations, 165 (2000), 315-354. doi: 10.1006/jdeq.2000.3780.  Google Scholar [14] F.-M. Huang and Y.-P. 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.  Google Scholar [15] F.-M. 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.  Google Scholar [16] F.-M. Huang, M. Mei, Y. Wang and T. Yang, Long-time behavior of solutions for bipolar hydrodynamic model of semiconductors with boundary effects, SIAM J. Math. Anal., 44, (2012), 1134-1164. doi: 10.1137/110831647.  Google Scholar [17] F.-M. Huang, M. Mei, Y. Wang and H. Yu, Asymptotic convergence to stationary waves for unipolar hydrodynami model of semiconductors, SIAM J. Math. Anal., 43 (2011), 411-429. doi: 10.1137/100793025.  Google Scholar [18] F.-M. 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.  Google Scholar [19] J. W. Jerome, Steady Euler-Poisson system: a differential/integral equation formulation with general constitutive relations, Nonlinear Anal., 71 (2009), e2188-e2193. doi: 10.1016/j.na.2009.04.042.  Google Scholar [20] A. Jüngel, "Quasi-hydrodynamic Semiconductor Equations," Progress in Nonlinear Differential Equations and their Applications, Vol 41, Birkhäuser Verlag, Besel-Boston-Berlin, 2001.  Google Scholar [21] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of solutions of the hydrodynamic model of semiconductors, Proc. Royal Soc. Edinburgh, Sect. A, 132 (2002), 359-378. doi: 10.1017/S0308210500001670.  Google Scholar [22] H.-L. Li, P. Markowich and M. Mei, Asymptotic behavior of subsonic entropy solutions of the isentropic Euler-Poisson equations, Quart. Appl. Math., 60 (2002), 773-796.  Google Scholar [23] C.-K. Lin, C.-T. Lin and M. Mei, Asymptotic behavior of solution to nonlinear damped p-system with boundary effect, Int. J. Numer. Anal. Model. Ser. B, 1 (2010), 70-92.  Google Scholar [24] T. Luo, R. Natalini and Z. Xin, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math., 59 (1998), 810-830. doi: 10.1.1.55.4600.  Google Scholar [25] P. Marcati, M. Mei and B. Rubino, Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping, J. Math. Fluid Mech. 7 (2005), suppl. 2, S224-S240 doi: 10.1007/s00021-005-0155-9.  Google Scholar [26] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal., 129 (1995), 129-145. doi: 10.1007/BF00379918.  Google Scholar [27] P. A. Markowich, C. A. Ringhofer and C. Schmeiser, "Semiconductor Equations," Springer-Verlag, Vienna, 1990.  Google Scholar [28] M. Mei, Best asymptotic profile for hyperbolic $p$-sytem with damping, SIAM J. Math. Anal., 42 (2010), 1-23. doi: 10.1137/090756594.  Google Scholar [29] M. Mei and Y. Wang, Stability of stationary waves for full Euler-Poisson system in multi-dimensional space, Commun. Pure Appl. Anal., 11 (2012), 1775-1807. doi: 10.3934/cpaa.2012.11.1775.  Google Scholar [30] R. Natalini, The bipolar hydrodynamic model for semiconductors and the drift-diffusion equations, J. Math. Anal. Appl., 198 (1996), 262-281. doi: 10.1006/jmaa.1996.0081.  Google Scholar [31] S. Nishibata and M. Suzuki, Asymptotic stability of a stationary solution to a hydrodynamic model of semiconductors, Osaka J. Math., 44 (2007), 639-665. doi: 10.1007/BF01210792.  Google Scholar [32] 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.  Google Scholar [33] F. Poupaud, M. Rascle and J.-P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential Equations, 123 (1995), 93-121. doi: 10.1006/jdeq.1995.1158.  Google Scholar [34] A. Sitenko and V. Malnev, "Plasma Physics Theory," Applied Mathematics and Mathematical Computation, 10. Chapman $&$ Hall, London, 1995.  Google Scholar [35] N. Tsuge, Existence and uniqueness of stationary solutions to one-dimensional bipolar hydrodynamic model of semiconductors, Nonlinear Analysis, 73 (2010), 779-787. doi: 10.1016/j.na.2010.04.015.  Google Scholar [36] B. Zhang, Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Comm. Math. Phys., 157 (1993), 1-22. doi: 10.1007/BF02098016.  Google Scholar [37] 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), 1-32. doi: 10.1006/jdeq.2000.3799.  Google Scholar [38] C. Zhu and H. Hattori, Asymptotic behavior of the solution to a nonisentropic hydrodynamic model of semiconductors, J. Differential Equations, 144 (1998), 353-389. doi: 10.1006/jdeq.1997.3381.  Google Scholar
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