Daniel Balagué, José Cañizo, Pierre Gabriel

We are concerned with the long-time behavior of the growth-fragmentation 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 [Caceres, Canizo, Mischler 2011, JMPA], 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.

Journal reference: Kinetic and Related Models, Vol. 6, No. 2, pp. 219-243 (2013)
arXiv Submitted on 28 Mar 2012 (v1), last revised 19 Feb 2013 (v2)