On the Global Existence and Blowup Phenomena of Schr\"{o}dinger Equations with Multiple Nonlinearities

TL;DR

This paper investigates the conditions under which solutions to a nonlinear Schrödinger equation with multiple nonlinearities exist globally or blow up, providing sharp thresholds and extending previous results in the field.

Contribution

It introduces new sufficient conditions and sharp thresholds for global existence and blowup in Schrödinger equations with combined nonlinearities, extending and improving prior work.

Findings

Established sharp thresholds for blowup and global existence.

Extended previous results to more general potentials and nonlinearities.

Provided conditions supplement existing theoretical frameworks.

Abstract

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=\Delta u-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0, &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. These results can be looked as the supplement to Chapter 6 of \cite{Cazenave2}. In addition, our results extend those of \cite{Zhang} and improve some of \cite{Tao2}.