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 \(B^{1}_{\infty ,1}(\mathbb {R})\). 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 \(B^{1}_{\infty ,1}(\mathbb {R})\).

中文翻译：

---

临界贝索夫空间中浅水波模型的短时解的适定性和对初始数据的非均匀依赖性

研究了包含著名的 Degasperis-Procesi 和 Fornberg-Whitham 方程的非线性浅水波浪方程。建立短时解的适定性，说明方程的解图在临界贝索夫空间\(B^{1}_{\infty ,1}(\mathbb {R})\)中是连续的。利用构造高频函数和低频函数的方法，我们证明了方程的解图是非均匀连续依赖于\(B^{1}_{\infty ,1}(\mathbb {R })\)。