On spectral gaps of growth-fragmentation semigroups in higher moment spaces

Mustapha Mokhtar-Kharroubi; Jacek Banasiak

doi:10.3934/krm.2021050

Article Contents

2022, Volume 15, Issue 2: 147-185. Doi: 10.3934/krm.2021050

This issue Previous Article Next Article Kinetic equations for processes on co-evolving networks

On spectral gaps of growth-fragmentation semigroups in higher moment spaces

Mustapha Mokhtar-Kharroubi^1, , and
Jacek Banasiak^2,

1.
Laboratoire de Mathématiques, CNRS-UMR 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon, France
2.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa, Institute of Mathematics, Łódź University of Technology, Łódź, Poland

^*Corresponding author: Mustapha Mokhtar-Kharroubi
^*Corresponding author: Mustapha Mokhtar-Kharroubi

Received: April 2021

Revised: November 2021

Early access: January 2022

Published: April 2022

Both authors were supported by DSI/NRF SARChI Grant 82770. The second author was also supported by the National Science Centre of Poland Grant 2017/25/B/ST1/00051

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Abstract

We present a general approach to proving the existence of spectral gaps and asynchronous exponential growth for growth-fragmentation semigroups in moment spaces $ L^{1}( \mathbb{R} _{+};\ x^{\alpha }dx) $ and $ L^{1}( \mathbb{R} _{+};\ \left( 1+x\right) ^{\alpha }dx) $ for unbounded total fragmentation rates and continuous growth rates $ r(.) $ such that $ \int_{0}^{+\infty } \frac{1}{r(\tau )}d\tau = +\infty .\ $ The analysis is based on weak compactness tools and Frobenius theory of positive operators and holds provided that $ \alpha >\widehat{\alpha } $ for a suitable threshold $ \widehat{ \alpha }\geq 1 $ that depends on the moment space we consider. A systematic functional analytic construction is provided. Various examples of fragmentation kernels illustrating the theory are given and an open problem is mentioned.

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Mathematics Subject Classification: Primary: 47D06, 47G20; Secondary: 47B65, 47A55.

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[1]	A. S. Ackleh and B. G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: Analysis and computations, J. Math. Biol., 35 (1997), 480-502. doi: 10.1007/s002850050062.
[2]	J. Banasiak, Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3319-3334. doi: 10.3934/dcdss.2020161.
[3]	J. Banasiak, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci., 14 (2004), 483-501. doi: 10.1142/S0218202504003325.
[4]	J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London, 2006.
[5]	J. Banasiak and W. Lamb, Growth-fragmentation-coagulation equations with unbounded coagulation kernels, Philos. Trans. Roy. Soc. A, 378 (2020), 20190612, 22 pp. doi: 10.1098/rsta.2019.0612.
[6]	J. Banasiak, W. Lamb and P. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, Volume 1, CRC Press, 2019.
[7]	J. Banasiak, L. O. Joel and S. Shindin, Long term dynamics of the discrete growth-decay-fragmentation equation, J. Evol. Equ., 19 (2019), 771-802. doi: 10.1007/s00028-019-00499-4.
[8]	J. Banasiak, K. Pichór and R. Rudnicki, Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166. doi: 10.1007/s10440-011-9666-y.
[9]	E. Bernard and P. Gabriel, Asymptotic behavior of the growth-fragmentation equation with bounded fragmentation rate, J. Funct. Anal, 272 (2017), 3455-3485. doi: 10.1016/j.jfa.2017.01.009.
[10]	E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, J. Evol. Equ, 20 (2020), 375-401. doi: 10.1007/s00028-019-00526-4.
[11]	J. Bertoin and A. R. Watson, A probabilistic approach to spectral analysis of growth-fragmentation equations, J. Funct. Anal., 274 (2018), 2163-2204. doi: 10.1016/j.jfa.2018.01.014.
[12]	W. Biedrzycka and M. Tyran-Kamińska, Self–similar solutions of fragmentation equations revisited, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 13-27. doi: 10.3934/dcdsb.2018002.
[13]	J. A. Cañizo, P. Gabriel and H. Yoldasz, Spectral gap for the growth-fragmentation equation via Harris's theorem, SIAM J. Math. Anal., 53 (2021), 5185-5214. doi: 10.1137/20M1338654.
[14]	W. Desch, Perturbations of Positive Semigroups in AL-Spaces, Unpublished manuscript, (1988).
[15]	O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme, On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248. doi: 10.1007/BF00277748.
[16]	M. Doumic Jauffret and P. Gabriel, Eigenelements of a general agregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X.
[17]	K.-J. Engel and R. Nagel, A Short Course on Operator Semigroups, Universitext. Springer, New York, 2006.
[18]	M. L. Greer, L. Pujo-Menjouet and G. F. Webb, A mathematical analysis of the dynamics of prion proliferation, J. Theor. Biol., 242 (2006), 598-606. doi: 10.1016/j.jtbi.2006.04.010.
[19]	T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math., 6 (1958), 261-322. doi: 10.1007/BF02790238.
[20]	P. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510. doi: 10.4310/CMS.2009.v7.n2.a12.
[21]	I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM Journal on Appl Math, 19 (1970), 607-628. doi: 10.1137/0119060.
[22]	P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, J. Math. Pures Appl., 84 (2005), 1235-1260. doi: 10.1016/j.matpur.2005.04.001.
[23]	S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898. doi: 10.1016/j.anihpc.2015.01.007.
[24]	M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on Advances in Mathematics for Applied Sciences, 46. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.
[25]	M. Mokhtar-Kharroubi, On the convex compactness property for the strong operator topology and related topics, Math. Methods Appl. Sci., 27 (2004), 687-701. doi: 10.1002/mma.497.
[26]	M. Mokhtar-Kharroubi, Compactness Properties of Perturbed Sub-Stochastic $C_{0}$-semigroups on $L^{1}(\mu )$ with Applications to Discreteness and Spectral gaps, Mém. Soc. Math. Fr, 2016.
[27]	M. Mokhtar-Kharroubi, On spectral gaps of growth-fragmentation semigroups with mass loss or death, Communications on Pure and Applied Analysis, (to appear), https://hal.archives-ouvertes.fr/hal-02962550/document.
[28]	M. Mokhtar-Kharroubi, Work inpreparation.
[29]	R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, vol. 1184, Springer-Verlag Berlin, 1986. doi: 10.1007/BFb0074922.
[30]	J. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory: Advances and Applications 88, Birkhäuser Verlag, Basel, 1996. doi: 10.1007/978-3-0348-9206-3.
[31]	A. Okubo, Dynamical aspects of animal grouping: Swarms, schools, flocks, and herds, Advances in Biophysics, 22 (1986), 1-94. doi: 10.1016/0065-227X(86)90003-1.
[32]	B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177. doi: 10.1016/j.jde.2004.10.018.
[33]	G. Schluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263—266. doi: 10.1007/BF01444620.
[34]	J. Voigt, Positivity in time dependent linear transport theory, Acta Appl. Math., 2 (1984), 311-331. doi: 10.1007/BF02280857.
[35]	J. Voigt, On resolvent positive operators and positive $C_{0}$-semigroups on AL-spaces, Semigroup Forum, 38 (1989), 263-266. doi: 10.1007/BF02573236.
[36]	L. Weis, A short proof for the stability theorem for positive semigroups on $L^{p}(\mu )$, Proc. Amer. Math. Soc, 126 (1998), 3253-3256. doi: 10.1090/S0002-9939-98-04612-7.