On blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

Abstract

We study the Cauchy problem for the nonlinear dispersive wave equation with dissipative term on the real line, ut-uxxt+<0 that includes the corresponding dissipative version of the Camassa-Holm equation as well as the hyperelastic-rod wave equation with dissipative term (f(u)=ku2/2 and g(u)= as special cases. In the present paper we demonstrate the simple and new conditions on the initial data that lead to blow-up of the solution. In particular, we establish local in space criterion (i.e., a criterion involving only the properties of the data in a neighborhood of a single point) which guarantees that solutions blow up in finite time.