Well-posedness and blow-up phenomena for a higher order shallow water equation

This paper deals with the Cauchy problem for a higher order shallow water equation y(t) + au(x)y + buy(x) = 0, where y := Lambda(2k)u equivalent to (I - partial derivative(2)(x))(k)u and k = 2. The local well-posedness of solutions for the Cauchy problem in Sobolev space H-s(R) with s >= 7/2 is obtained. Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown, and conditions that lead to the development of singularities in finite time for the solutions are also acquired. Finally, the weak solution for the equation is considered. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.