Strichartz estimates for Schr\"odinger equations with nonlinear boundary interactions

Abstract

We study a Schr\"odinger equation in the upper half-space with a nonlinear Neumann boundary interaction driven by the Bessel operator $\Ba$, $a>-1$. The problem arises naturally as an extension formulation for a nonlocal NLS with memory and can also be interpreted as a Schr\"odinger evolution with a nonlinear singular source concentrated on a codimension-one interface. We first develop a complete linear theory for the associated inhomogeneous problem with nonhomogeneous Neumann data. A central ingredient is a new Duhamel representation formula that separates bulk and boundary dynamics and identifies the precise role of the boundary propagator. Using this formula, we establish sharp Strichartz estimates adapted to the geometry of the half-space and the singular structure induced by the Bessel operator. The analysis reveals a basic dichotomy between the regimes $a\ge 0$ and $-1<a<0$: in the former, bulk and boundary exhibit a unified dispersive behavior, whereas in the latter the dispersive structure becomes anomalous, requiring weighted estimates and distinct functional frameworks for the bulk and boundary contributions. As an application of the linear theory, we prove well-posedness results for the nonlinear problem in the $L_a^2$-mass critical and subcritical regimes. For $a\ge 0$, we obtain global well-posedness for sufficiently small critical data, together with local and global results in the subcritical case. In the anomalous range $-1<a<0$, we establish existence and uniqueness on arbitrary finite time intervals for sufficiently small critical data, as well as local well-posedness in the subcritical regime. The results provide a unified dispersive framework for Schr\"odinger equations with nonlinear boundary interactions and singular extension structures associated with nonlocal-in-time dynamics.