Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate

Abstract

The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.

References in corpus (12)

• Eigenelements of a General Aggregation-Fragmentation Model
• On the long time behavior of the TCP window size process
• Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations
• A probabilistic approach to spectral analysis of growth-fragmentation equations
• Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
• Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates
• Probabilistic aspects of critical growth-fragmentation equations
• Substochastic semigroups and densities of piecewise deterministic Markov processes
• Rate of convergence to self-similarity for the fragmentation equation in L^1 spaces
• Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate
• Exponential relaxation to self-similarity for the superquadratic fragmentation equation
• Self-similar solutions of fragmentation equations revisited

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• A non-conservative Harris ergodic theorem
• Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation
• The strong Malthusian behavior of growth-fragmentation processes
• Growth--fragmentation--coagulation equations with unbounded coagulation kernels
• On spectral gaps of growth-fragmentation semigroups in higher moment spaces
• Explicit solutions to some fragmentation equations with growth or decay
• Strong laws of large numbers for a growth-fragmentation process with bounded cell sizes