We consider Smoluchowski's equation with a homogeneous kernel of the form with and . We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at in the case . We also give some partial uniqueness results for self-similar profiles: in the case we prove that two profiles with the same mass and moment of order are necessarily equal, while in the case we prove that two profiles with the same moments of order and , and which are asymptotic at , are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.
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José A. Cañizo, Stéphane Mischler, Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski’s coagulation equation. Rev. Mat. Iberoam. 27 (2011), no. 3, pp. 803–839DOI 10.4171/RMI/653