On the decaying property of quintic NLS on 3D hyperbolic space

Abstract

In this paper, we study the (pointwise) decaying property of quintic NLS on the three-dimensional hyperbolic space $H^3$. We show the nonlinear solution enjoys the same decay rate as the linear solution does. This result is based on the associated global well-posedness and scattering result in Ionescu et al. (2012). This extends (Fan and Zhao, 2021)’ Euclidean works to the hyperbolic space with additional improvements in regularity requirement (lower and almost critical regularity assumed). Realizing such improvements also work for the Euclidean case, we obtain a result for the fourth-order NLS analogue studied in Yu et al. (2023) recently with better, i.e. almost critical regularity assumption.