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

Abstract

The study of low regularity Cauchy data for nonlinear dispersive PDEs has been successfully achieved using modulation spaces in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS) iu<inf>t</inf>+Δu±|x|−b|u|αu=0(b,α>0) on the whole space Rn having initial data in modulation spaces. In the subcritical regime (0<α<4−2bn), we establish local well-posedness in L2+Mα+2,α+2α+1(⊃L2+Hsfors>nα2(α+2)). By adapting Bourgain's high-low decomposition method, we establish global well-posedness in Mp,pp−1 with 2<p and p sufficiently close to 2. This is the first global well-posedness result for INLS in modulation spaces, which contains certain Sobolev Hs(0<s<1) and L<inf>s</inf>p−Sobolev spaces. © 2026 Elsevier Inc.