April  2022, 15(2): 147-185. doi: 10.3934/krm.2021050

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

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

Received  April 2021 Revised  November 2021 Published  April 2022 Early access  January 2022

Fund Project: 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

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.

Citation: Mustapha Mokhtar-Kharroubi, Jacek Banasiak. On spectral gaps of growth-fragmentation semigroups in higher moment spaces. Kinetic and Related Models, 2022, 15 (2) : 147-185. doi: 10.3934/krm.2021050
References:
[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. BanasiakL. 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. BanasiakK. 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. CañizoP. 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. DiekmannH. 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. GreerL. 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. MichelS. 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.

show all references

References:
[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. BanasiakL. 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. BanasiakK. 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. CañizoP. 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. DiekmannH. 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. GreerL. 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. MichelS. 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.

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