Sharp local well-posedness of KdV type equations with dissipative perturbations

TL;DR

This paper establishes sharp local well-posedness results for KdV type equations with dissipative perturbations, focusing on initial data in $L^2$-based Sobolev spaces using bilinear estimates in weighted spaces.

Contribution

It provides the first sharp local well-posedness results for dissipative perturbations of KdV equations in $L^2$-based Sobolev spaces.

Findings

Derived bilinear estimates in weighted time spaces

Established sharp local well-posedness results

Analyzed linear perturbations of KdV equations

Abstract

In this work, we study the initial value problems associated to some linear perturbations of KdV equations. Our focus is in the well-posedness issues for initial data given in the $L^2$-based Sobolev spaces. We derive bilinear estimate in a space with weight in the time variable and obtain sharp local well-posedness results.