We prove that if $u_1,,u_2$ are solutions of the Benjamin-Ono equation defined in $ (x,t)inR times [0,T]$ which agree in an open set $Omegasubset R times [0,T]$, then $u_1equiv u_2$. We extend this uniqueness result to a general class of equations of Benjamin-Ono type in both the initial value problem and the initial periodic boundary value problem. This class of 1-dimensional non-local models includes the intermediate long wave equation. Finally, we present a slightly stronger version of our uniqueness results for the Benjamin-Ono equation.
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