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Persistence properties and unique continuation of solutions of the Camassa-Holm equation. (English) Zbl 1142.35078

The authors study the Camassa-Holm equation

${u}_{t}-{u}_{txx}+3u{u}_{x}-2{u}_{x}{u}_{xx}-u{u}_{xxx}=0,$

which physically was derived as a shallow water equation admitting peaked solitons. They prove that, given a strong solution to the Cauchy problem for this equation such that the initial data $u\left(x,0\right)$ decays exponentially together with the spatial derivative, if at some time ${t}_{1}>0$ the solution $u\left(x,{t}_{1}\right)$ decays exponentially in $x$ then $u$ is identically zero: $u\left(x,t\right)\equiv 0$.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q35 PDEs in connection with fluid mechanics 35B60 Continuation of solutions of PDE
##### Keywords:
Camassa-Holm equation; Cauchy problem
##### References:
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