Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations

María J. Cáceres, José A. Cañizo, Stéphane Mischler

Research output: Contribution to journalArticleResearchpeer-review

28 Citations (Scopus)

Abstract

We study the asymptotic behavior of linear evolution equations of the type ∂tg=Dg+Lg-λg, where L is the fragmentation operator, D is a differential operator, and λ is the largest eigenvalue of the operator Dg+Lg. In the case Dg=-∂xg, this equation is a rescaling of the growth-fragmentation equation, a model for cellular growth; in the case Dg=-∂x(xg), it is known that λ=1 and the equation is the self-similar fragmentation equation, closely related to the self-similar behavior of solutions of the fragmentation equation ∂tf=Lf.By means of entropy-entropy dissipation inequalities, we give general conditions for g to converge exponentially fast to the steady state G of the linear evolution equation, suitably normalized. In other words, the linear operator has a spectral gap in the natural L2 space associated to the steady state. We extend this spectral gap to larger spaces using a recent technique based on a decomposition of the operator in a dissipative part and a regularizing part. © 2011 Elsevier Masson SAS.
Original languageEnglish
Pages (from-to)334-362
JournalJournal des Mathematiques Pures et Appliquees
Volume96
Issue number4
DOIs
Publication statusPublished - 1 Oct 2011

Keywords

  • Entropy
  • Exponential convergence
  • Fragmentation
  • Growth
  • Long-time behavior
  • Self-similarity

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