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 \begin{document}$ L^{1} $\end{document} spaces. We prove first generation of \begin{document}$ C_{0} $\end{document} -semigroups governing them for unbounded total fragmentation rate and fragmentation kernel \begin{document}$ b(.,.) $\end{document} such that \begin{document}$ \int_{0}^{y}xb(x,y)dx = y-\eta (y)y $\end{document} ( \begin{document}$ 0\leq \eta (y)\leq 1 $\end{document} expresses the mass loss) and continuous growth rate \begin{document}$ r(.) $\end{document} such that \begin{document}$ \int_{0}^{\infty }\frac{1}{r(\tau )}d\tau = +\infty . $\end{document} This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in \begin{document}$ L^{1} $\end{document} 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.