Quantum (fluid) Equations Self-similarity Analysis

In this paper, we mainly consider the global existence of the solutions, large time behavior ofwhereρ, n are electron density and hole density, E is the electric field,the pressure functions p(ρ) =ρ^m,p(n) = n^m(m > 1),ε> 0 is the scaled Planck constant which is small in general andρ±are given constants. Letwhere W is the unique self-similar solution ofρ_t= (ρ^m)_xx (see [19])and x₀ satisfiesOur main result as follows:Theorem Assume thatρ₀ > 0, n₀ > 0, (g₀, h₀)∈H³Ã—H³,‖g₀‖_H³(R)+‖h₀‖_H³(R)small enough and |ρ₊ -ρ_-|<< 1,then there is unique global classical solution(ρ, n,E) of the Cauchy problem (1), such that where C,βare positive constants.