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

Abstract

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, u(t) - u(xxt) + [f (u)](x) - [f (u)](xxx) + [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) = ku(2) /2 and g (u) (3-kappa) 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 up in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.(C) 2017 Elsevier Inc. All rights reserved.