Unique Continuation Properties for solutions to the Camassa-Holm equation and other non-local equations

Abstract

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.

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References in corpus (2)

• Lipschitz metric for the Camassa-Holm equation on the line
• Global conservative solutions of the Camassa-Holm equation for initial data nonvanishing asymptotics