非齐次Schrdinger方程的整体解与爆破解

Global Existence and Blow-up for Inhomogeneous Schrdinger Equation

研究如下非齐次Schrdinger方程itΦ=-ΔΦ-|x|-b|Φ|p-1Φx∈R^n,t≥0,0<b<2,n≥3当1<p<1+4-2b/n或者p=1+4-2b/n且初始质量充分小时,得到其Cauchy问题在H1(Rn)中整体适定;当1+4-2b/n≤p<1+4-2b/n-2时,得到其Cauchy问题的解在有限时间爆破的充分条件.

The aim of this paper is to study the cauchy problem of the following inhomogeneous Schrdinger equation itΦ =-ΔΦ-｜x｜-b｜Φ｜p-1Φ x ∈ R^n,t≥0,0b2,n≥3 When 1p1＋4-2b/n,the global well-posedness in H^1（R^n）has been established;when p=1＋4-2b/n,a mass critical is derived for the global well-posedness in H^1（R^n）;when 1＋4-2b/n≤p1＋4-2b/n-2,we have obtained the finite-time blow-up of solution under certain conditions.