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武汉大学学报 英文版 | Wuhan University Journal of Natural Sciences
Wan Fang
Wuhan University
Latest Article
Asymptotic Behavior of Solutions of the Bipolar Quantum Drift-Diffusion Model in the Quarter Plane
LIU fang, LI Yeping
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
In this study, we consider the one-dimensional bipolar quantum drift-diffusion model, which consists of the coupled nonlinear fourth-order parabolic equation and the electric field equation. We first show the global existence of the strong solution of the initial boundary value problem in the quarter plane. Moreover, we show the self-similarity property of the strong solution of the bipolar quantum drift-diffusion model in the large time. Namely, we show the unique global strong solution with strictly positive density to the initial boundary value problem of the quantum drift-diffusion model, which in large time, tends to have a self-similar wave at an algebraic time-decay rate. We prove them in an energy method.
Key words:asymptotic behavior; quantum drift-diffusion model; self-similar wave; energy estimate
CLC number:O 175.2
[1]	Jüngel A. Quasi-hydrodynamic Semiconductor Equations. Part of the Progress in Nonlinear Differential Equations[M]. Basel: Birkhäuser, 2001.
[2]	Jüngel A, Matthes D. A derivation of the isothermal quantum hydrodynamic equations using entropy minimization [J]. ZAMM—Journal of Applied Mathematics and Mechanics, 2005, 85(11): 806-814.
[3]	Gardner C L. The quantum hydrodynamic model for semi-conductor devices [J]. Siam Journal on Applied Mathematics, 2007, 54(2): 409-427.
[4]	Ancona M G, Iafrate G J. Quantum correction to the equation of state of an electron gas in a semiconductor [J]. Phys Rev B, 1989, 39(13): 9536-9540.
[5]	Jüngel A. A positivity preserving numerical scheme for a nonlinear fourth order parabolic system [J]. Siam Journal on Numerical Analysis, 2002, 39(2): 385-406.
[6]	Odanaka S . Multidimensional discretization of the stationary quantum drift-diffusion model for ultrasmall  MOSFET structures [J]. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 2004, 23(6): 837-842. 
[7]	Jerome J. Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices [M]. Heidelberg : Springer- Verlag, 1996.
[8]	Markowich P A, Ringhofev C A , Schmeiser C. Semicon-ductor Equations [M]. New York: Springer-Verlag, 1990: 5171-5188.
[9]	Abdallah N B, Unterreiter A. On the stationatry quatum drift diffusion model [J]. Z Angew Math Phys, 1998, 49: 251-275. 
[10]	Jüngel A, Pinnau R. Global nonnegative solutions of a non-linear fourth-order parabolic equation for quantum systems [J]. SIAM J Math Anal, 2000, 32(14): 760-777.
[11]	Chen L, Ju Q C. Existence of weak solution and semiclassical limit for quantum drift-diffusion model [J]. Z Angew Math Phys, 2007, 58(11): 1-15.
[12]	Chen X Q, Chen L, Jian H Y. The existence and long-time behavior of weak solution to bipolar quantum drift-diffusion model [J]. Chin Ann Math, 2007, 28(6): 651-664.
[13]	Ju Q C, Chen L. Semiclassical limit for biporlar quantum drift-diffusion model [J]. Acta Mathematica Sinica—English Series, 2009, 29(12): 285-293.
[14]	Jüngel A, Violet I. The quasineutral limit in the quantum drift-diffusion equations [J]. Asymptotic Analysis, 2007, 53 (3): 139-157.
[15]	Chen X Q, Chen L. The bipolar quantum drift-diffusion model [J]. Acta Math Sinica, 2009, 25(4): 617-638. 
[16]	Liu Y N, Sun W L, Li Y P. Existence of global attractor for the one-dimensional bipolar quantum drift-diffusion model [J]. Wuhan Univ J Nat Sci, 2017, 22(3): 227-282.
[17]	Li H L, Zhang G J, Zhang M, et al. Long-time self-similar asymptotic of the macroscopic quantum models [J]. J Math Phys, 2008, 49(7): 073503.
[18]	Blesher P M, Lebowitz J L, Speer E R. Existence and posi-tivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations [J]. Comm Pure Appl Math, 1994, 47(7): 923-942.
[19]	Gianazza U, Toscani G. The Wasserstein gradient flow of the Fisher information and the quantum drift-diffusion equation [J]. Arch Ration Mech Anal, 2009, 194(9): 133-220.
[20]	Li Y P. Long-time self-similarity of classical solutions to the bipolar quantum hydrodynamic models [J]. Nonlinear Anal-ysis, 2001, 74(4): 1501-1512.
[21]	Li Y P. Global existence and large time behavior of solutions for the bipolar quantum hydrodynamic models in the quarter plane [J]. Math Meth Appl Sci, 2013, 36(11): 1409- 1422.
[22]	Marcati P, Mei M, Rubino B. Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping [J]. J Math Fluid Mech, 2005, 7(2): S224- S240.
[23]	Nishibata S, Suzuki M. Initial boundary value problems for a quantum hydrodynamic model of semiconductors: Asymptotic behaviors and classical limits [J]. J Differential Equations, 2008, 244(4): 836-874.
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