On a class of solutions to the generalized KdV type equation

TL;DR

This paper establishes local well-posedness and regularity propagation for solutions to a generalized KdV equation with low non-linearity degree, expanding understanding of solution behavior in weighted Sobolev spaces.

Contribution

It introduces a novel approach to prove local well-posedness for low-degree non-linear generalized KdV equations and confirms the propagation of regularity for these solutions.

Findings

Proved local well-posedness in weighted Sobolev spaces.

Demonstrated propagation of regularity in solutions.

Extended existing results to equations with low non-linearity.

Abstract

We consider the IVP associated to the generalized KdV equation with low degree of non-linearity \begin{equation*} \partial_t u + \partial_x^3 u \pm |u|^{\alpha}\partial_x u = 0,\; x,t \in \mathbb{R},\;\alpha \in (0,1). \end{equation*} By using an argument similar to that introduced by Cazenave and Naumkin [2] we establish the local well-posedness for a class of data in an appropriate weighted Sobolev space. Also, we show that the solutions obtained satisfy the propagation of regularity principle proven in [3] in solutions of the $k$-generalized KdV equation.