On the well-posedness of the Cauchy problem for the generalized Korteweg-de Vries-Burgers equation

TL;DR

This paper establishes sharp well-posedness and ill-posedness results for the generalized Korteweg-de Vries-Burgers equation in Sobolev spaces, depending on the regularity index and the parameter alpha.

Contribution

It provides the first precise thresholds for well-posedness and ill-posedness of the equation in Sobolev spaces, extending understanding of the equation's behavior.

Findings

Well-posedness for s > -min{(3+2α)/4, 1}

Ill-posedness for s < -min{(3+2α)/4, 1} when 1/2 ≤ α ≤ 1

Sharp thresholds for solution regularity in Sobolev spaces

Abstract

Considered is the generalized Korteweg-de Vries-Burgers equation $$ u_{t}+u_{xxx}+uu_{x}+|D_{x}|^{2\alpha}u=0,\quad t\in \mathbb{R}^{+}, x\in \mathbb{R}, $$ with $0\leq \alpha\le 1$. We prove a sharp results on the associated Cauchy problem in the Sobolev space $ {H}^s(\mathbb{R})$. For $s>-\min\{\frac {3+2\alpha}4, 1\}$ we give the well-posedness of solutions of the Cauchy problem, while for $\frac 12\le\alpha\le 1$ and for $s<-\min\{\frac {3+2\alpha}4, 1\}$ we show some ill-posedness issues.