On spectral gaps of growth-fragmentation semigroups in higher moment spaces

Abstract

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $L^{1}(\mathbb{R}_{+};\ x^{\alpha }dx)$ and $L^{1}(\mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx)$ for unbounded total fragmentation rates and continuous growth rates $r(.)$\ such that $\int_{0}^{+\infty } \frac{1}{r(\tau )}d\tau =+\infty .\ $The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $\alpha >\widehat{\alpha }$ for a suitable threshold $\widehat{ \alpha }\geq 1$ that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.