Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS

TL;DR

This paper investigates the existence, uniqueness, and classification of minimal mass blow-up solutions for a 2D inhomogeneous focusing mass-critical nonlinear Schrödinger equation, extending previous results to non-constant inhomogeneities.

Contribution

It establishes necessary and sufficient conditions on the inhomogeneity for critical mass blow-up solutions and classifies minimal blow-up solutions at non-degenerate points.

Findings

Identifies conditions on $k(x)$ for existence of blow-up solutions.

Classifies minimal blow-up solutions at non-degenerate points.

Extends Merle's classification to inhomogeneous cases.

Abstract

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+\Delta u+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2}<M_k$ are global in time while a finite time blow up singularity formation may occur for $\|u\|_{L^2}>M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.