Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation

TL;DR

This paper constructs minimal blow-up solutions for the mass-critical inhomogeneous nonlinear Schrödinger equation, revealing conditions under which solutions blow up at a critical point with maximal rate, including on certain surfaces.

Contribution

It introduces a method to find minimal blow-up solutions in the inhomogeneous setting, extending understanding of blow-up dynamics in mass-critical NLS with external potentials.

Findings

Existence of finite-time blow-up solutions at critical points with flat inhomogeneity.

Minimal mass blow-up solutions occur at maximum points of the inhomogeneity.

Application to unstable blow-up solutions on specific surfaces.

Abstract

We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrodinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations.