Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

Pierre Gabriel (1)

1 LJLL - Laboratoire Jacques-Louis Lions

We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Thanks the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can ''asymptotically'' be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved thanks to the Poincaré-Bendixon theorem or by Hopf bifurcation.

Mots clés: size-structured populations; relative entropy; long-time asymptotics; self-similarity; eigenproblem; growth-fragmentation processes; stability; periodic solution; Poincaré-Bendixon theorem; Hopf bifurcation

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Gabriel. Long-time Asymptotics for Nonlinear Growth-fragmentation Equations. Communications in Mathematical Sciences, 2012, 10 (3), pp.787-820. ⟨10.4310/CMS.2012.v10.n3.a4⟩. ⟨hal-00565863v2⟩