Global solutions and blow-up phenomena to a shallow water equation (Q984438)

A nonlinear shallow water equation, which includes the famous Camassa-Holm (CH) and Degasperis-Procesi (DP) equations as special cases, is investigated. By applying the Kato theorem the local well-posedness of solutions for the nonlinear equation in the Sobolev space \(H^s(\mathbb R)\) with \(s>3/2\) is developed. Provided that \((1-\partial_\chi^2)u_0\) does not change sign, \(u_0\in H^s\) \((s>3/2)\) and \(u_0\in L^1(\mathbb R)\), the existence and uniqueness of the global solutions to the equation are shown to be true in \(C([0,\infty);H^s(\mathbb R))\cap C^1([0,\infty);H^{s-1}(\mathbb R))\). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.