Well-posedness and continuity properties of the Fornberg–Whitham equation in the Besov space \(B^1_{\infty ,1}(\mathbb {R})\)

For the Fornberg–Whitham equation, the local well-posedness in the critical Besov space \(B_{p, 1}^{1+\frac{1}{p}}(\mathbb {R})\) $B_{p,1}^{1+\frac{1}{p}}(\mathbb{R})$ with \(1\le p <\infty \) $1\le p<\infty$ has been studied in Guo (Nonlinear Anal RWA 70:103791, 2023). However, for the endpoint case \(p=\infty \) $p=\infty$ , whether it is locally well-posed or ill-posed in \(B_{\infty , 1}^{1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ is still open. In this paper, we prove that the Fornberg–Whitham equation is well-posed in the critical Besov space \(B_{\infty , 1}^{1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ with solutions depending continuously on initial data, which is different from that of the Camassa–Holm equation (Guo et al. in J Differ Equ 327:127, 2022). In addition, we show that this dependence is sharp by showing that the solution map is not uniformly continuous on the initial data.