Global well-posedness and scattering in weighted space for nonlinear Schrödinger equations below the Strauss exponent without gauge-invariance

TL;DR

This work tackles global well-posedness and scattering for the nonlinear Schrödinger equation with general homogeneous nonlinearities in dimensions up to three, including non-gauge-invariant terms and exponents at and below the Strauss threshold. A key contribution is the development of a new framework based on a pseudo-conformal transform and a resolvent-based decomposition, which yields an $H^1$-level control in a fixed-point setting despite the lack of time-continuity in weighted formulations. The analysis leverages detailed operator bounds for resolvents, dilation- and conjugation-based identities, and a careful decomposition into $A_{j,n}$ terms whose nonlinear interactions are summable under weighted coefficient assumptions. The results extend small-data global existence and scattering to a broad class of homogeneous nonlinearities in weighted spaces, including the intercritical range in two dimensions, and establish small-data scattering in $H^{1,1}$, with implications for understanding long-time behavior beyond gauge-invariant models.

Abstract

In this paper, we consider the nonlinear Schrödinger equation (NLS) with a general homogeneous nonlinearity in dimensions up to three. We assume that the degree (i.e., power) of the nonlinearity is such that the equation is mass-subcritical and short-range. We establish global well-posedness (GWP) and scattering for small data in the standard weighted space for a class of homogeneous nonlinearities, including non-gauge-invariant ones. Additionally, we include the case where the degree is less than or equal to the Strauss exponent. When the nonlinearity is not gauge-invariant, the standard Duhamel formulation fails to work effectively in the weighted Sobolev space; for instance, the Duhamel term may not be well-defined as a Bochner integral. To address this issue, we introduce an alternative formulation that allows us to establish GWP and scattering, even in the presence of poor time continuity of the Duhamel term.

Theorems & Definitions (10)

• proof : Proof of Lemma \ref{['L2/16']}
• proof : Proof
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• proof : Proof
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• proof : Proof of Proposition \ref{['P:key']}
• proof : Proof of Corollary \ref{['C:both']}