Low-regularity global solution of the inhomogeneous nonlinear Schr\"odinger equations in modulation spaces

Abstract

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.