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

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 \(\sigma>3+\frac{1}{p},\ 1\leq p\leq+\infty\).