Spectral analysis of semigroups and growth-fragmentation equations

Abstract

The aim of this paper is twofold:

(1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of classical results such as the spectral mapping theorem, some (quantified) Weyl's Theorems and the Krein–Rutman Theorem. Motivated by evolution PDE applications, the results apply to a wide and natural class of generators which split as a dissipative part plus a more regular part, without assuming any symmetric structure on the operators nor Hilbert structure on the space, and give some growth estimates and spectral gap estimates for the associated semigroup. The approach relies on some factorization and summation arguments reminiscent of the Dyson–Phillips series in the spirit of those used in [96,87,51,86].

(2) On the other hand, we present the semigroup spectral analysis for three important classes of “growth-fragmentation” equations, namely the cell division equation, the self-similar fragmentation equation and the McKendrick–Von Foerster age structured population equation. By showing that these models lie in the class of equations for which our general semigroup analysis theory applies, we prove the exponential rate of convergence of the solutions to the associated first eigenfunction or self-similar profile for a very large and natural class of fragmentation rates. Our results generalize similar estimates obtained in [103,73] for the cell division model with (almost) constant total fragmentation rate and in [19,18] for the self-similar fragmentation equation and the cell division equation restricted to smooth and positive fragmentation rate and total fragmentation rate which does not increase more rapidly than quadratically. It also improves the convergence results without rate obtained in [84,36] which have been established under similar assumptions to those made in the present work.