Well-posedness of short time solutions and non-uniform dependence on the initial data for a shallow water wave model in critical Besov space

A nonlinear shallow water wave equation containing the famous Degasperis-Procesi and Fornberg-Whitham equations is investigated. The well-posedness of short time solutions is established to illustrate that the solution map of the equation is continuous in the critical Besov space \(B^{1}_{\infty ,1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ . Using the methods to construct high and low frequency functions, we prove that the solution map of the equation is non-uniform continuous dependence on the initial data in \(B^{1}_{\infty ,1}(\mathbb {R})\) $B_{\infty ,1}^1(\mathbb{R})$ .