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The Existence and Long-Time Behavior of Weak Solution to Bipolar Quantum Drift-Diffusion Model*

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Abstract

The authors study the existence and long-time behavior of weak solutions to the bipolar transient quantum drift-diffusion model, a fourth order parabolic system. Using semi-discretization in time and entropy estimate, the authors get the global existence of nonnegative weak solutions to the one-dimensional model with nonnegative initial and homogenous Neumann (or periodic) boundary conditions. Furthermore, by a logarithmic Sobolev inequality, it is proved that the periodic weak solution exponentially approaches its mean value as time increases to infinity.

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Author information

Correspondence to Xiuqing Chen.

Additional information

* Project supported by the National Natural Science Foundation of China (Nos. 10631020, 10401019) and the Basic Research Grant of Tsinghua University.

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Chen, X., Chen, L. & Jian, H. The Existence and Long-Time Behavior of Weak Solution to Bipolar Quantum Drift-Diffusion Model*. Chin. Ann. Math. Ser. B 28, 651–664 (2007). https://doi.org/10.1007/s11401-006-0568-7

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Keywords

  • Quantum drift-diffusion
  • Weak solution
  • Long-time behavior

2000 MR Subject Classification

  • 35k35
  • 35J60
  • 65M12
  • 65M20