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 $\alpha >\hat{\alpha }$ for a suitable threshold $\hat{\alpha }\ge 1$ 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.

39 pages

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