The local and global existence of solutions for a generalized Camassa-Holm equation

A fully nonlinear generalization of the Camassa-Holm equation is investigated. Using the pseudoparabolic regularization technique, its local well-posedness in Sobolev space H ^s (?) with $s > \tfrac{3} {2}$ is established via a limiting procedure. Provided that the initial momentum (1 -? _x ² )u ₀ satisfies the sign condition, u ₀ ∈ H ^s (?) $\left( {s > \tfrac{3} {2}} \right)$ and u ₀ ε L ₁(?), the existence and uniqueness of global solutions for the equation are shown to be true in the space C(0,∞);H ^s (?)) ∩ C ¹(0,g8);H ^s?1(?)).