Remarks on strong global solutions of the [formula omitted]-equation

Abstract

A bounded first-order derivative with respect to x of a solution to the b-equation, which encompasses the Camassa–Holm and Degasperis–Procesi equations, serves as a crucial condition that triggers a process leading to the establishment of bounds for derivatives of the solution up to an order determined by the regularity of the initial data. Consequently, this condition ensures the global existence of strong solutions in Sobolev spaces.