Wave breaking and propagation speed for the Camassa–Holm equation with κ≠0κ≠0

In this paper, we consider the Camassa–Holm equation with κ≠0κ≠0 on the real line. We establish certain conditions on the initial datum to guarantee that the corresponding solution exists globally or blows up in finite time. Infinite propagation speed is proved in the following sense: the corresponding solution u(x,t)+κu(x,t)+κ with compactly supported initial datum (u0(x)+κ∈C0∞(R)) does not   have compact x−x−support in its lifespan.