Unique continuation properties for solutions to the Camassa-Holm equation and related models

Abstract

It is shown that if u ( x , t ) u(x,t) is a real solution of the initial value problem for the Camassa-Holm equation which vanishes in an open set Ω ⊂ R × [ 0 , T ] \Omega \subset \mathbb {R}\times [0,T] , then u ( x , t ) = 0 , ( x , t ) ∈ R × [ 0 , T ] u(x,t)=0,(x,t)\in \mathbb {R}\times [0,T] . 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. This result also applies to solutions of the initial periodic boundary value problems associated to these models.