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 $B_{p,r}^s,s>1+\frac{1}{p},1\le p,r\le +\infty$ and critical Besov spaces $B_{p,1}^{1+\frac{1}{p}},1\le p<+\infty$, 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 $B_{p,r}^s,s>1+\frac{1}{p},1\le p\le +\infty ,1\le r<+\infty$ and critical Besov spaces $B_{p,1}^{1+\frac{1}{p}},1\le p<+\infty$. At last, we show that the solution is ill-posed in $B_{p,\infty }^{\sigma }$ with $\sigma >3+\frac{1}{p},1\le p\le +\infty$.

References in corpus (2)

• The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations
• Ill-posedness for the Camassa-Holm and related equations in Besov spaces