## Small Data Scattering for the Global Solutions of the Supercritical Generalized KdV Equation

Xiaoya Tang *

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR, China.

Shanshan Zheng

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, PR, China.

*Author to whom correspondence should be addressed.

### Abstract

We consider the scattering problem for the global solutions of the supercritical generalized KdV equation *∂ _{t}u+∂_{xxx}u+μ∂_{x}*(

*u*

^{k+1}) = 0, where

*k*> 4 is an integer, initial data

*u*

_{0 }belongs to

*H*

^{1}(

**ℝ**), and

*μ*= ±1. To solve the scattering problem, a scattering criteria is established firstly, and then a new inequality is introduced to obtain uniformly bounded solutions in

*H*

^{1}(

**ℝ**). Finally, we further clarify the conditions for the equation to have a global solution scattering in

*H*

^{1}(

**ℝ**). Our method is mainly inspired by the works of Farah, Linares, Pastor, and Visciglia.

Keywords: Supercritical, generalized KdV equation, scattering, global solution

#### How to Cite

*Asian Journal of Mathematics and Computer Research*,

*31*(1), 26–41. https://doi.org/10.56557/ajomcor/2024/v31i18519

### Downloads

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