Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

TL;DR

This paper extends the analysis of blow-up solutions in the intercritical inhomogeneous nonlinear Schrödinger equation from radial to non-radial initial data, providing bounds on blow-up rates and concentration phenomena.

Contribution

It generalizes previous radial results to non-radial cases, establishing bounds on blow-up rates and concentration for finite time blow-up solutions.

Findings

Existence of blow-up solutions in non-radial case

Upper and lower bounds for blow-up rates

Concentration phenomena for blow-up solutions

Abstract

In this paper we consider the inhomogeneous nonlinear Schr\"odinger (INLS) equation \begin{align}\label{inls} i \partial_t u +\Delta u +|x|^{-b} |u|^{2\sigma}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2\sigma}$ satisfies $0<s_c<1$. In a previous work, for radial initial data in $\dot H^{s_c}\cap \dot H^1$, we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in $H^1$.