Long-time self-similar asymptotic of the macroscopic quantum models

Details Long-time self-similar asymptotic of the macroscopic quantum models Long-time self-similar asymptotic of the macroscopic quantum models

2008-11-24

arxiv.org/abs/0811.3786v1

J. Math. Phys., 49(07), 073503, 14pp, 2008

Physics

Mathematical Physics

18 pages

Scientific paper

10.1063/1.2949082

The unipolar and bipolar macroscopic quantum models derived recently for instance in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.

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