Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

Abstract

The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted \(L^1\) spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.

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Correspondence to Pierre Gabriel.

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The Pierre Gabriel was supported by the ANR Project KIBORD, ANR-13-BS01-0004, Funded by the French Ministry of Research.

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Bernard, É., Gabriel, P. Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate. J. Evol. Equ. 20, 375–401 (2020). https://doi.org/10.1007/s00028-019-00526-4

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Keywords

  • Growth-fragmentation equation
  • Uniform asynchronous exponential growth
  • Positive semigroups
  • Quasi-compactness
  • Creation of moments

Mathematics Subject Classification

  • 35B40 (primary)
  • 35P05
  • 35Q92
  • 35R09
  • 47D06 (secondary)