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
is developed. Provided that
does not change sign,
, the existence and uniqueness of the global solutions to the equation are shown to be true in
. Conditions that lead to the development of singularities in finite time for the solutions are also acquired.