# American Institute of Mathematical Sciences

June  2013, 6(2): 219-243. doi: 10.3934/krm.2013.6.219

## 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, Spain 2 School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom 3 Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex, France

Received  October 2012 Revised  November 2012 Published  February 2013

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.
Citation: Daniel Balagué, José A. Cañizo, Pierre Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. Kinetic & Related Models, 2013, 6 (2) : 219-243. doi: 10.3934/krm.2013.6.219
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##### References:
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