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Self-Similar Solutions to Coagulation Equations with Time-Dependent Tails: The Case of Homogeneity One

  • Marco BonaciniEmail author
  • Barbara Niethammer
  • Juan J. L. Velázquez
Article
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Abstract

We prove the existence of a one-parameter family of self-similar solutions with time dependent tails for Smoluchowski’s coagulation equation, for a class of kernels \({K(x,y)}\) which are homogeneous of degree one and satisfy \({K(x,1) \to k_0 > 0}\) as \({x\to 0}\). In particular, we establish the existence of a critical \({\rho_* > 0}\) with the property that for all \({\rho\in(0,\rho_*)}\) there is a positive and differentiable self-similar solution with finite mass M and decay \({A(t)x^{-(2+\rho)}}\) as \({x\to\infty}\), with \({A(t)=e^{M(1+\rho)t}}\). Furthermore, we show that (weak) self-similar solutions in the class of positive measures cannot exist for large values of the parameter \({\rho}\).

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Acknowledgements

The authors acknowledge support from the CRC 1060 The mathematics of emergent effects at the University of Bonn that is funded through the German Science Foundation (DFG).

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Marco Bonacini
    • 1
    Email author
  • Barbara Niethammer
    • 1
  • Juan J. L. Velázquez
    • 1
  1. 1.Institut für Angewandte MathematikUniversität BonnBonnGermany

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