Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

Abstract

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.

Cited by in corpus (9)

• A non-conservative Harris ergodic theorem
• Asymptotic behaviour of neuron population models structured by elapsed-time
• Self-similarity in a General Aggregation-Fragmentation Problem ; Application to Fitness Analysis
• Exponential relaxation to self-similarity for the superquadratic fragmentation equation
• Global stability for the prion equation with general incidence
• Optimization of an Amplification Protocol for Misfolded Proteins by using Relaxed Control
• A meeting point of entropy and bifurcations in cross-diffusion herding
• Derivation and mathematical study of a sorption-coagulation equation
• Global stability in a competitive infection-age structured model