  It is shown that if $\,u(x,t)\,$ is a solution of the initial value problem
for the Camassa-Holm equation which vanishes in an open set $\,\Omega\subset
\mathbb R\times [0,T]$, then $\,u(x,t)=0,\,(x,t)\in\mathbb R\times [0,T]$. This
result also applies to solutions of the initial periodic boundary value
problems associated to the Camassa-Holm equation. The argument of proof can be
placed in a general setting to extend the above results to a class of
non-linear non-local 1-dimensional models which includes the Degasperis-Procesi
equation.
