Kinetic and Related Models (KRM)

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

Pages: 219 - 243, Volume 6, Issue 2, June 2013      doi:10.3934/krm.2013.6.219

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Daniel Balagué - Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain (email)
José A. Cañizo - School of Mathematics, Watson Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom (email)
Pierre Gabriel - Laboratoire de Mathématiques de Versailles, CNRS UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, 45 Avenue de États-Unis, 78035 Versailles cedex, France (email)

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.

Keywords:  Fragmentation, growth, eigenvalue problem, entropy, exponential convergence, long-time behavior.
Mathematics Subject Classification:  Primary: 35B40, 45C05, 45K05; Secondary: 82D60, 92C37.

Received: October 2012;      Revised: November 2012;      Available Online: February 2013.