Persistence Properties and Unique Continuation of solutions of the   Camassa-Holm equation
  It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution. 
A. Alexandrou Himonas, Gerard Misio{\l}ek, Gustavo Ponce, and Yong   Zhou