Global existence of weak solutions for 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. Provided that initial value u0 ∈ Hs (1 ≤ s ≤ 3 2 ), u0 ∈ L1(R) and (1 − ∂2 x )u0 does not change sign, it is shown that there exists a unique global weak solution to the equation.