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 yt+auxy+buyx=0yt+auxy+buyx=0, where y:=Λ2ku≡(I−∂x2)ku and k=2k=2. The local well-posedness of solutions for the Cauchy problem in Sobolev space Hs(R)Hs(R) with s⩾7/2s⩾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.

► We model a higher order shallow water equation which presents fine structural properties.
► Under some assumptions, the existence and uniqueness of the global solutions to the equation are shown.
► The conditions that lead to the development of singularities in finite time for the solutions are also acquired.