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

Abstract

<p style='text-indent:20px;'>We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in <inline-formula><tex-math id="M1">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula> spaces. We prove first generation of <inline-formula><tex-math id="M2">\begin{document}$ C_{0} $\end{document}</tex-math></inline-formula>-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel <inline-formula><tex-math id="M3">\begin{document}$ b(.,.) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M4">\begin{document}$ \int_{0}^{y}xb(x,y)dx = y-\eta (y)y $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M5">\begin{document}$ 0\leq \eta (y)\leq 1 $\end{document}</tex-math></inline-formula> expresses the mass loss) and continuous growth rate <inline-formula><tex-math id="M6">\begin{document}$ r(.) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M7">\begin{document}$ \int_{0}^{\infty }\frac{1}{r(\tau )}d\tau = +\infty . $\end{document}</tex-math></inline-formula>This is done in the spaces of finite mass or finite mass and number of agregates. Generation relies on unbounded perturbation theory peculiar to positive semigroups in <inline-formula><tex-math id="M8">\begin{document}$ L^{1} $\end{document}</tex-math></inline-formula> 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.</p>