Wave breaking and propagation speed for a class of nonlocal dispersive θθ-equations

In this paper we investigate a class of nonlocal dispersive models—the θθ-equations derived recently by Liu (2008) [1]. These new kinds of equations admit a blow-up phenomenon and infinite propagation speed like the Camassa–Holm equation and the Degasperis–Procesi equation. First, for the special structure of these equations, we establish sufficient conditions on the initial data for guaranteeing the formulation of a singularity in the sense that the derivative of the solution blows up in finite time. It turns out that it is the shape rather than the size and smoothness of the initial data that influences the breakdown in finite time. Moreover, infinite propagation speed for the θθ-equations is proved in the following sense: the corresponding solution u(x,t)u(x,t) with compactly supported initial datum u0(x)u0(x) does not   have compact xx-support any longer in its lifespan. Finally, we show that these kinds of equations have unique continuation properties for the strong solution.