Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
121 tokens/sec
GPT-4o
9 tokens/sec
Gemini 2.5 Pro Pro
47 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Remarks on Scattering Properties of the Solution to a Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities (1009.0325v2)

Published 2 Sep 2010 in math.AP

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.

Summary

We haven't generated a summary for this paper yet.