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We prove that the peakons are asymptotically H 1-stable, under the flow ofthe Degasperis-Procesi equation, in the class of functions with a momentumdensity that belongs to M + (R). The key argument is a rigidity result foruniformly in time exponentially decaying global solutions that is shared by theHolm-Staley b-family of equations for b $\ge$ 1. This extends previous resultsobtained for the Camassa-Holm equation (b = 2).

A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon