Fine  asymptotics of profiles and relaxation to equilibrium for  growth-fragmentation equations with variable drift rates

Daniel Balagué; José A. Cañizo; Pierre Gabriel

doi:10.3934/krm.2013.6.219

Article Contents

2013, Volume 6, Issue 2: 219-243. Doi: 10.3934/krm.2013.6.219

This issue Previous Article Next Article Large deviations for the solution of a Kac-type kinetic equation

Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra
2.
School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham B15 2TT
3.
Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex

Received: September 30, 2012

Revised: October 31, 2012

Published: May 2013

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Abstract

We are concerned with the long-time behavior of the growth-frag-mentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to $0$ and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [1], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.

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Mathematics Subject Classification: Primary: 35B40, 45C05, 45K05; Secondary: 82D60, 92C37.

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