LAUSR

Abstract This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, u t − u x x t + [ f ( u ) ] x − [ f ( u ) ] x x x + [ g ( u ) + f ″ ( u ) 2 u x 2 ] x = 0 that includes the Camassa–Holm equation as well as the hyperelastic-rod wave equation ( f ( u ) = k u 2 / 2 and g ( u ) = ( 3 − k ) u 2 / 2 ) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u 0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.