Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

International audienceWe 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