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

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 $B^s_{p,r}, s>1+frac{1}{p}, 1leq p,rleq+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$, 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 $B^s_{p,r}, s>1+frac{1}{p}, 1leq pleq+infty, 1leq r<+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$. At last, we show that the solution is ill-posed in $B^{sigma}_{p,infty}$ with $sigma>3+frac{1}{p}, 1leq pleq+infty$.