Global existence and blow-up for the variable coefficient Schr\"{o}dinger equations with a linear potential

TL;DR

This paper investigates the conditions for global existence and blow-up of solutions to variable coefficient Schrödinger equations with a linear potential, considering radial, finite variance, and nonradial cases with various nonlinearities.

Contribution

It extends the analysis of Schrödinger equations by establishing global existence and blow-up criteria for variable coefficient cases with linear potential, including nonradial initial data.

Findings

Global existence below ground state threshold for radial and finite variance cases.

Blow-up results for supercritical nonlinearities.

Sufficient conditions for global behavior in nonradial cases.

Abstract

In this paper, we study a class of variable coefficient Schr\"{o}dinger equations with a linear potential \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,\] where $2-n<b\leq0,\ c\geq b-2$ and $0<\textbf{p}_c\leq(2-b)(p+2)$, where $\textbf{p}_c:=np-2c$. In the radial or finite variance case, we firstly prove the global existence and blow-up below the ground state threshold for the mass-critical and inter-critical nonlinearities. Next, adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a sufficient condition on the nonradial initial data, under which the global behavior of the general solution is established.