The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{ p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$iu_t + \Delta u\pm |x|^{ -b}|u|^{ \alpha}u=0,$$ where $\alpha, b>0,$ on whole space $\mathbb R^n$ in modulation spaces. In the subcritical regime $(0<\alpha< \frac{ 4-2b}{ n}),$ we establish local well-posedness in $L^{ 2}+M^{ \alpha+2,\frac{ \alpha+2}{ \alpha+1}}( \supset L^2 + H^s \ \text{ for} \ s>\frac{ n\alpha}{ 2(\alpha+2)}).$ By adapting Bourgain's high-low decomposition method, we establish global well-posedness in $M^{ p,\frac{ p}{ p-1}}$ with $2<p$ and $p$ sufficiently close to 2. This is the first global well-posedness result for INLS on modulation spaces, which contains certain Sobolev $H^s$ $(0<s<1)$ and $L^p_s-$Sobolev spaces.
Categories: math.AP