On well-posedness for some Korteweg-De Vries type equations with variable coefficients

TL;DR

This paper establishes well-posedness results for variable coefficient KdV-type equations, extending previous work by lowering regularity requirements and incorporating dispersive estimates with a novel approach.

Contribution

It proves existence, uniqueness, and local well-posedness for KdV equations with variable coefficients under new conditions, using a combined change of variables and dispersive estimates.

Findings

Proved existence and uniqueness of solutions in Sobolev spaces.

Established local well-posedness in $H^s( )$ for $s>1/2$.

Extended previous results to lower regularity settings.

Abstract

In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(\mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(\mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.