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

Daniel Balagué (1) , José Cañizo (1) , Pierre Gabriel (2)

1 UAB - Departament de Matemàtiques [Barcelona]
2 BEAGLE - Artificial Evolution and Computational Biology

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.

Mots clés: Fragmentation; growth; eigenvalue problem; entropy; exponential convergence; long-time behavior

Domaines: Equations aux dérivées partielles [math.AP]

Daniel Balagué, José Cañizo, Pierre Gabriel. Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates. 2012. ⟨hal-00683148v1⟩