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

A nonlinear shallow water equation, which includes the famous Camassa–Holm (CH) and Degasperis–Procesi (DP) equations as special cases, is investigated. The local well-posedness of solutions for the nonlinear equation in the Sobolev space Hs(R) with is developed. Provided that does not change sign, u0 Hs ( ) and u0 L1(R), the existence and uniqueness of the global solutions to the equation are shown to be true in u(t,x) C([0,∞);Hs(R))∩C1([0,∞);Hs−1(R)). Conditions that lead to the development of singularities in finite time for the solutions are also acquired.