On spectral gaps of growth-fragmentation semigroups with mass loss or death.

We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in L¹ spaces. We prove first generation of C₀-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel b (. ,.) such that ∫₀^y xb(x, y)dx = y − η(y)y (0 ≤ η (y) ≤ 1 expresses the mass loss) and continuous growth rate r (.) such that ∫₀∞ 1/r(τ) dτ = + ∞. This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in L¹ spaces. Secondly, we show that the semigroup has a spectral gap and asynchronous exponential growth. The analysis relies on weak compactness tools and Frobenius theory of positive operators. A systematic functional analytic construction is provided. [ABSTRACT FROM AUTHOR]