  We unify a few of the best known results on wave breaking for the
Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J.
Hyman and others) in a single theorem: a sufficient condition for the breakdown
is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the
real line. Such blowup criterion looks more natural than the previous ones, as
the condition on the initial data is purely local in the space variable. Our
method relies on the introduction of two families of Lyapunov functions.
Contrary to McKean's necessary and sufficient condition for blowup, our
approach applies to other equations that are not integrable: we illustrate this
fact by establishing new local-in-space blowup criteria for an equation
modeling nonlinear dispersive waves in elastic rods.
