Global existence and blow-up solutions for a nonlinear shallow water equation

• DOI: 10.1007/s00208-006-0768-1
• Publication Date: 2006-05-03T13:07:03Z
• Authors: Yue Liu
• Keywords: 0101 mathematics; 01 natural sciences

Considered herein are the problems of the existence of global solutions and the formation of singularities for a new nonlinear shallow water wave equation derived by Dullin, Gottward and Holm. Blow-up can occur only in the form of wave-breaking. A wave-breaking mechanism for solutions with certain initial profiles is described in detail and the exact blow-up rate is established. The blow-up set for a class of initial profiles and lower bounds of the existence time of the solution are also determined.