Threshold solutions for the $3d$ cubic INLS: the energy-subcritical case
Luccas Campos, Luiz Gustavo Farah, Jason Murphy
TL;DR
This work extends the threshold-dynamics classification for the 3D cubic INLS equation $i\partial_t u + \Delta u + |x|^{-b}|u|^2 u = 0$ from the previously treated range $b\in(0,\tfrac{1}{2})$ to the full energy-subcritical interval $b\in(0,1)$. The authors replace pointwise decay estimates with Strichartz-type control of the modulation remainder $g$ and derive a modulation-parameter stability analysis that avoids the need for $\nabla\Delta Q\in L^2$, enabling sharper control of the parameter $\alpha$ and the deviation $\delta(t)$. Their main contributions are (i) establishing $L_t^pL_x^q$ bounds for $g$ and (ii) proving uniform decay $|\alpha(s)-\alpha(t)|\lesssim e^{-ct}$, which in turn yields $\delta(t)\lesssim e^{-ct}$ and convergence of $u(t)$ to the ground-state orbit $e^{i(\zeta_0+t)}Q$ as $t\to\infty$, thereby completing the threshold classification for all $b\in(0,1)$. The approach leverages Strichartz estimates and a refined modulation-virial framework, with potential applicability to other dispersive equations featuring singular potentials.
Abstract
We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schrödinger equations of the form $i\partial_t u + Δu + |x|^{-b}|u|^2 u = 0$ in the range $b\in(0,\tfrac12)$. By modifying the modulation analysis and using Strichartz estimates in place of pointwise bounds, we extend the result to the full energy-subcritical range $b\in(0,1)$. This strategy is expected to carry over to other dispersive equations with singular potentials.