Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces

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 Schrödinger equation (INLS) $$i{u}_t+\mathrm{\Delta }u\pm \mathrm{\mid }x{\mathrm{\mid }}^{-b}\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0,$$ 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 $L^2+M^{\alpha +2,\frac{\alpha +2}{\alpha +1}}(\supset L^2+H^s\text{ for }s>\frac{n\alpha }{2(\alpha +2)})\mathrm{.}$ 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 $L_s^p-$Sobolev spaces.