Long-time asymptotics for nonlinear growth-fragmentation equations

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. Using 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 the existence of periodic solutions is proved with the Poincaré-Bendixon theorem or by Hopf bifurcation.

Keywords

size-structured populations, growth-fragmentation processes, eigenproblem, selfsimilarity, relative entropy, long-time behavior, stability, periodic solution, Poincaré-Bendixon theorem, Hopf bifurcation

2010 Mathematics Subject Classification

35B10, 35B32, 35B35, 35B40, 35B42, 35Q92, 37G15, 45K05, 92D25

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Published 9 April 2012