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Spectral analysis of semigroups and growth-fragmentation equations. (English) Zbl 1357.47044
Authors’ 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 [C. Mouhot, Commun. Math. Phys. 261, No. 3, 629–672 (2006; Zbl 1113.82062); the first author and C. Mouhot, ibid. 288, No. 2, 431–502 (2009; Zbl 1178.82056); “Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation”, Preprint, arXiv:1412.7487; M. P. Gualdani et al., “Factorization of non-symmetric operators and exponential \(H\)-theorem”, Preprint, arXiv:1006.5523].
(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 [B. Perthame and L. Ryzhik, J. Differ. Equations 210, No. 1, 155–177 (2005; Zbl 1072.35195); P. Laurençot and B. Perthame, Commun. Math. Sci. 7, No. 2, 503–510 (2009; Zbl 1183.35038)] for the cell division model with (almost) constant total fragmentation rate and in [M. J. Cáceres et al., J. Math. Pures Appl. (9) 96, No. 4, 334–362 (2011; Zbl 1235.35034); Commun. Appl. Ind. Math. 1, No. 2, Article ID 590, 299–308 (2010; Zbl 1329.82064)] 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 [P. Michel et al., J. Math. Pures Appl. (9) 84, No. 9, 1235–1260 (2005; Zbl 1085.35042); M. Escobedo et al., Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22, No. 1, 99–125 (2005; Zbl 1130.35025)] which have been established under similar assumptions to those made in the present work.

MSC:
47D06 One-parameter semigroups and linear evolution equations
35P15 Estimates of eigenvalues in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
92D25 Population dynamics (general)
34G10 Linear differential equations in abstract spaces
34K30 Functional-differential equations in abstract spaces
35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
45C05 Eigenvalue problems for integral equations
45K05 Integro-partial differential equations
92C37 Cell biology
82D60 Statistical mechanics of polymers
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