The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces

TL;DR

This paper investigates the well-posedness and ill-posedness of the Fornberg-Whitham equation in various Besov spaces, establishing local solutions, their dependence on initial data, and conditions leading to ill-posedness.

Contribution

The paper improves existing results by proving local well-posedness in broader Besov spaces and identifying spaces where solutions are not uniformly continuous or ill-posed.

Findings

Established local well-posedness in supercritical and critical Besov spaces.

Proved solutions lack uniform continuous dependence on initial data in certain Besov spaces.

Showed ill-posedness in Besov spaces with higher regularity.

Abstract

In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p,r\leq+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$, which improves the previous work \cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p\leq+\infty,\ 1\leq r<+\infty$ and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$. At last, we show that the solution is ill-posed in $B^{\sigma}_{p,\infty}$ with $\sigma>3+\frac{1}{p},\ 1\leq p\leq+\infty$.