On solutions to the Holm–Staley b-family of equations

In this paper we consider a family of one-dimensional shallow water equations (the Holm–Staley b-family of equations) derived recently by Holm and Staley (2003 Phys. Lett. A 308 437–44). Analogous to the Camassa–Holm equation, these new equations admit blow-up phenomenon and infinite propagation speed. First, we establish blow-up results for this family of equations under various classes of initial data. It turns out that it is the shape instead of the size and smoothness of the initial data which influences breakdown in finite time. Then, infinite propagation speed for the Holm–Staley b-family of equations is proved in the following sense: the corresponding solution u(x, t) with compactly supported initial datum u₀(x) does not have a compact x-support any longer in its lifespan. Moreover, we show that for any fixed time t > 0 in its lifespan, the corresponding solution u(x, t) behaves as u(x, t) = L(t)e^−x for x ≫ 1 and u(x, t) = l(t)e^x for x ≪ −1, with a strictly increasing function L(t) > 0 and a strictly decreasing function l(t) < 0, respectively.