Remarks on Scattering Properties of the Solution to a Nonlinear Schr\"{o}dinger Equation with Combined Power-Type Nonlinearities

TL;DR

This paper investigates the scattering behavior of solutions to a nonlinear Schrödinger equation with combined power nonlinearities, addressing open problems and revealing new scattering properties through analysis of nonlinear interactions.

Contribution

It introduces novel analysis of the interaction between different nonlinear terms in the Schrödinger equation, partially solving open problems and expanding understanding of scattering phenomena.

Findings

Partial solutions to open problems in nonlinear Schrödinger equations

New scattering properties identified for combined nonlinearities

Analysis of nonlinear interaction effects on solution behavior

Abstract

In this paper, we consider the Cauchy problem of Nonlinear Schr\"{o}dinger equation \begin{align*} \left\{\begin{array}{ll}&i u_t+\Delta u=\lambda_1|u|^{p_1}u+\lambda_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=\varphi(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $N\geq 3$, $0<p_1<p_2<\frac{4}{N-2}$, $\lambda_1$ and $\lambda_2$ are real constants. Using the methods in \cite{Cazenave2} and analyzing the interaction between the nonlinearity $\lambda_1|u|^{p_1}u$ and $\lambda_2|u|^{p_2}u$, we not only partly solve the open problems of Terence Tao, Monica Visan and Xiaoyi Zhang's \cite{Tao} but also obtain other scattering properties of the solutions.