Minimal-mass blow-up solutions for inhomogeneous nonlinear Schr\"{o}dinger equations with growth potentials

TL;DR

This paper constructs a finite-time blow-up solution with minimal mass for an inhomogeneous nonlinear Schrödinger equation, extending previous results to less restrictive conditions on the potentials.

Contribution

It demonstrates the existence of critical-mass blow-up solutions under weaker assumptions on the potentials' smoothness and boundedness, including unbounded and non-Lipschitz cases.

Findings

Constructed a critical-mass blow-up solution.

Extended existence results to less smooth and unbounded potentials.

Described the solution's behavior near blow-up time.

Abstract

In this paper, we consider the following equation: \[ i\frac{\partial u}{\partial t}+\Delta u+g(x)|u|^{\frac{4}{N}}u-Wu=0. \] We construct a critical-mass solution that blows up at a finite time and describe the behaviour of the solution in the neighbourhood of the blow-up time. Banica-Carles-Duyckaertz (2011) has shown the existence of a critical-mass blow-up solution under the assumptions that $N\leq 2$, that $g$ and $W$ are sufficiently smooth and that each derivative of these is bounded. In this paper, we show the existence of a critical-mass blow-up solution under weaker assumptions regarding smoothness and boundedness of $g$ and $W$. In particular, it includes the cases where $W$ is growth at spatial infinity or not Lipschitz continuous.