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

TL;DR

This paper proves that macroscopic quantum models for charge transport in one-dimensional space exhibit self-similar behavior over long times, with solutions approaching a self-similar wave at an algebraic decay rate.

Contribution

It is the first to establish the self-similarity property and long-time asymptotics of these quantum models, including existence and uniqueness of solutions.

Findings

Existence of a unique global strong solution with positive density.

Solutions tend to a self-similar wave in large time.

Decay rate of solutions follows an algebraic pattern.

Abstract

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.