Local-in-space blow-up criteria for a class of nonlinear dispersive wave equations

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, ut−uxxt+[f(u)]x−[f(u)]xxx+[g(u)+f″(u)2ux2]x=0 that includes the Camassa-Holm equation as well as the hyperelastic-rod wave equation (f(u)=ku2/2 and g(u)=(3−k)u2/2) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.