The global weak solution for a generalized Camassa-Holm equation (Q1949520)

Summary: A nonlinear generalization of the famous Camassa-Holm model is investigated. Provided that initial value \(u_0 \in H^s(\mathbb{R})(1 \leq s \leq 3/2)\) and \((1 - \partial^2_x)u_0\) satisfies an associated sign condition, it is shown that there exists a unique global weak solution to the equation in space \(u(t, x) \in L^2([0, +\infty), H^s(\mathbb{R}))\) in the sense of distribution, and \(u_x \in L^\infty([0, +\infty) \times \mathbb{R})\).