Kinetic and Related Models doi:10.3934/krm.2013.6.801 c American Institute of Mathematical Sciences Volume 6, Number 4, December 2013 pp. 801–808 ON SELF-SIMILAR SOLUTIONS TO THE HOMOGENEOUS BOLTZMANN EQUATION Marco Cannone Universit´ e Paris-Est–Marne-la-Vall´ ee Laboratoire d’Analyse et de Math´ ematiques Appliqu´ ees, UMR 8050 CNRS 5 boulevard Descartes, Cit´ e Descartes Champs-sur-Marne 77454 Marne-la-Vall´ ee cedex 2, France Grzegorz Karch Instytut Matematyczny, Uniwersytet Wroclawski pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland This work is dedicated to the memory of Seiji Ukai Abstract. In this note, our results from [Comm. Pure Appl. Math. 63 (2010), 747–778] on infinite energy solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules are discussed, presented in a different context, and improved by using recent observations by Morimoto and Yang. In particular, similarities between the homogeneous Boltzmann equation and the fractional heat equation are emphasized. Moreover, we show that a certain conjecture by Bobylev and Cercignani on regularity of self-similar solutions to the homogeneous Boltzmann equation for Maxwellian-type molecules has a positive answer. 1. Introduction. Due to the celebrated Boltzmann H-theorem, it is natural to conjecture that solutions of the Boltzmann equation converge towards an equi- librium which is given by the Maxwellian distribution. A substantial progress in proving this conjecture was done in the case of several models from the collision ki- netic theory which are variants of the Boltzmann equation: e.g. the Fokker-Planck type equation, a model for granular media, the homogeneous Boltzmann equation. We refer the reader to the review article [17] for a discussion of those results and for several references. In this work, we will show that the convergence of solutions towards Maxwellians is not the only possible scenario in the study of the large time behavior of solutions to Boltzmann equations. Here, we consider the homogeneous Boltzmann equation in R 3 ∂ t f (v,t)= Q(f,f )(v,t) (1) 2010 Mathematics Subject Classification. Primary: 82C40; Secondary: 76P05. Key words and phrases. Homogeneous Boltzmann equation, Maxwellian gas, self-similar solu- tions, asymptotic stability. The work of G. Karch was partially supported by the MNiSzW grant No. N N201 418839 and the Foundation for Polish Science operated within the Innovative Economy Operational Pro- gramme 2007-2013 funded by European Regional Development Fund (Ph.D. Programme: Mathe- matical Methods in Natural Sciences). 801