Well-posedness and non-uniform dependence on initial data for the Fornberg–Whitham-type equation in Besov spaces

In this paper, we first establish the local well-posedness for the Fornberg–Whitham-type equation in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) $B_{p,r}^s(\mathbb{R})$ with \( 1\le p,r\le \infty \) $1\le p,r\le \infty$ and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\) $s>max\{1+\frac{1}{p},\frac{3}{2}\}$ , which improve the previous work in Sobolev spaces \( H^{s}({\mathbb {R}})= B^{s}_{2,2}({\mathbb {R}})\) $H^s(\mathbb{R})=B_{2,2}^s(\mathbb{R})$ with \( s>\frac{3}{2}\) $s>\frac{3}{2}$ (Lai and Luo in J Differ Equ 344:509–521, 2023). Furthermore, we prove the solution is not uniformly continuous dependence on the initial data in the Besov spaces \(B^{s}_{p,r}({\mathbb {R}})\) $B_{p,r}^s(\mathbb{R})$ with \( 1\le p\le \infty \) $1\le p\le \infty$ , \( 1\le r< \infty \) $1\le r<\infty$ and \(s> max\{1+\frac{1}{p},\frac{3}{2}\}\) $s>max\{1+\frac{1}{p},\frac{3}{2}\}$ .