April  2022, 21(4): 1293-1327. doi: 10.3934/cpaa.2022019

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

Laboratoire de Mathématiques, CNRS-UMR 6623, Université de Bourgogne Franche-Comté, 16 Route de Gray, 25030 Besançon, France

Received  June 2021 Revised  December 2021 Published  April 2022 Early access  January 2022

We give a general theory on well-posedness and time asymptotics for growth fragmentation equations in $ L^{1} $ spaces. We prove first generation of $ C_{0} $-semigroups governing them for unbounded total fragmentation rate and fragmentation kernel $ b(.,.) $ such that $ \int_{0}^{y}xb(x,y)dx = y-\eta (y)y $ ($ 0\leq \eta (y)\leq 1 $ expresses the mass loss) and continuous growth rate $ r(.) $ such that $ \int_{0}^{\infty }\frac{1}{r(\tau )}d\tau = +\infty . $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 $ L^{1} $ 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.

Citation: Mustapha Mokhtar-Kharroubi. On spectral gaps of growth-fragmentation semigroups with mass loss or death. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1293-1327. doi: 10.3934/cpaa.2022019
References:
[1]

O. Arino and R. Rudnicki, Stability of phytoplankton dynamics, C. R. Biol., 327 (2004), 961–969.

[2]

J. Banasiak, On conservativity and shattering for an equation of phytoplankton dynamics, C. R. Biol., 327 (2004), 1025-1036. 

[3]

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.

[4] J. BanasiakW. Lamb and Ph. Laurençot., Analytic Methods for Coagulation-Fragmentation Models, Vol Ⅰ, CRC Press, 2019. 
[5]

J. Banasiak and W. Lamb, Growth-fragmentation-coagulation equations with unbounded coagulation kernels, Phil. Trans. Roy. Soc. A, 378 (2020), 20190612. doi: 10.1098/rsta.2019.0612.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10] B. Davies, One-parameter Semigroups, Academic Press, 1980. 
[11]

W. Desch, Perturbations of positive semigroups in AL-spaces; unpublished manuscript, 1988.,

[12]

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.

[13]

K. J. Engel and R. Nagel, One-Parameter Semigroups for linear evolution equations, Graduate Texts in Mathematics 194, Springer-Verlag, 2000.

[14]

H. EnglerJ. Prüss and G. F. Webb, Analysis of a model for the dynamics of prions, II, J. Math. Anal. Appl., 324 (2006), 98-117.  doi: 10.1016/j.jmaa.2005.11.021.

[15]

P. Gabriel and M. Doumic Jauffret, Eigenelements of a general agregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[16]

J. HuangB. F Edwards and A. D Levine, General solutions and scaling violation for fragmentation with mass loss, J. Phys. A, 24 (1991), 3967-3977. 

[17]

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.

[18]

Ph. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/ Cell-Division equation, Commun. Math. Sci, 7 (2009), 503-510. 

[19]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.

[20]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer Lecture Notes in Biomathematics, Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.

[21]

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.

[22]

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.

[23]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on Adv in Math for Appl Sci, World Scientific, 1997. doi: 10.1142/9789812819833.

[24]

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.

[25]

M. Mokhtar-Kharroubi, On $L^{1}$ spectral theory of neutron transport, Differ. Integral Equ., 11 (2005), 1221-1242. 

[26]

M. Mokhtar-Kharroubi, On $L^{1}$ exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 266 (2014), 6418-6455.  doi: 10.1016/j.jfa.2014.03.019.

[27]

M. Mokhtar-Kharroubi and J. Voigt, On honesty of perturbed substochastic $C_{0}$-semigroups in $L^{1}$-spaces, J. Op. Th, 64 (2010), 131–147.

[28]

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), 80 pp.

[29]

M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete and Cont Dyn Syst, Series B, 23 (2018), 4087–4116. doi: 10.3934/dcdsb.2018127.

[30]

M. Mokhtar-Kharroubi and Q. Richard, Spectral theory and time asymptotics of size-structured two-phase population models, Discrete and Cont Dyn Syst, Series B, 25 (2020), 2969–3004. doi: 10.3934/dcdsb.2020048.

[31]

M. Mokhtar-Kharroubi, Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics, Discrete and Cont Dyn Syst, Series S, 13 (2020), 3551–3563. doi: 10.3934/dcdss.2020244.

[32]

M. Mokhtar-Kharroubi, On spectral gaps of growth-fragmentation semigroups with mass loss or death, Preprint, hal-02962550, 2020.

[33]

M. Mokhtar-Kharroubi and J. Banasiak, On spectral gaps of growth-fragmentation semigroups in higher moment spaces, Kinetic & Related Models, to appear.

[34]

R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Springer-Verlag Berlin, 1986. doi: 10.1007/BFb0074922.

[35]

B. Perthame, Transport Equations in Biology. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[36]

G. Schluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[37]

J. Voigt, On resolvent positive operators and positive $ C_{0}$-semigroups on AL-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.

[38]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.2307/2000695.

show all references

References:
[1]

O. Arino and R. Rudnicki, Stability of phytoplankton dynamics, C. R. Biol., 327 (2004), 961–969.

[2]

J. Banasiak, On conservativity and shattering for an equation of phytoplankton dynamics, C. R. Biol., 327 (2004), 1025-1036. 

[3]

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.

[4] J. BanasiakW. Lamb and Ph. Laurençot., Analytic Methods for Coagulation-Fragmentation Models, Vol Ⅰ, CRC Press, 2019. 
[5]

J. Banasiak and W. Lamb, Growth-fragmentation-coagulation equations with unbounded coagulation kernels, Phil. Trans. Roy. Soc. A, 378 (2020), 20190612. doi: 10.1098/rsta.2019.0612.

[6]

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.

[7]

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.

[8]

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.

[9]

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.

[10] B. Davies, One-parameter Semigroups, Academic Press, 1980. 
[11]

W. Desch, Perturbations of positive semigroups in AL-spaces; unpublished manuscript, 1988.,

[12]

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.

[13]

K. J. Engel and R. Nagel, One-Parameter Semigroups for linear evolution equations, Graduate Texts in Mathematics 194, Springer-Verlag, 2000.

[14]

H. EnglerJ. Prüss and G. F. Webb, Analysis of a model for the dynamics of prions, II, J. Math. Anal. Appl., 324 (2006), 98-117.  doi: 10.1016/j.jmaa.2005.11.021.

[15]

P. Gabriel and M. Doumic Jauffret, Eigenelements of a general agregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783. doi: 10.1142/S021820251000443X.

[16]

J. HuangB. F Edwards and A. D Levine, General solutions and scaling violation for fragmentation with mass loss, J. Phys. A, 24 (1991), 3967-3977. 

[17]

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.

[18]

Ph. Laurençot and B. Perthame, Exponential decay for the growth-fragmentation/ Cell-Division equation, Commun. Math. Sci, 7 (2009), 503-510. 

[19]

I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM J. Appl. Math., 19 (1970), 607-628.  doi: 10.1137/0119060.

[20]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer Lecture Notes in Biomathematics, Springer, New York, 1986. doi: 10.1007/978-3-642-93287-8_2.

[21]

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.

[22]

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.

[23]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory. New Aspects, Series on Adv in Math for Appl Sci, World Scientific, 1997. doi: 10.1142/9789812819833.

[24]

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.

[25]

M. Mokhtar-Kharroubi, On $L^{1}$ spectral theory of neutron transport, Differ. Integral Equ., 11 (2005), 1221-1242. 

[26]

M. Mokhtar-Kharroubi, On $L^{1}$ exponential trend to equilibrium for conservative linear kinetic equations on the torus, J. Funct. Anal., 266 (2014), 6418-6455.  doi: 10.1016/j.jfa.2014.03.019.

[27]

M. Mokhtar-Kharroubi and J. Voigt, On honesty of perturbed substochastic $C_{0}$-semigroups in $L^{1}$-spaces, J. Op. Th, 64 (2010), 131–147.

[28]

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), 80 pp.

[29]

M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete and Cont Dyn Syst, Series B, 23 (2018), 4087–4116. doi: 10.3934/dcdsb.2018127.

[30]

M. Mokhtar-Kharroubi and Q. Richard, Spectral theory and time asymptotics of size-structured two-phase population models, Discrete and Cont Dyn Syst, Series B, 25 (2020), 2969–3004. doi: 10.3934/dcdsb.2020048.

[31]

M. Mokhtar-Kharroubi, Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics, Discrete and Cont Dyn Syst, Series S, 13 (2020), 3551–3563. doi: 10.3934/dcdss.2020244.

[32]

M. Mokhtar-Kharroubi, On spectral gaps of growth-fragmentation semigroups with mass loss or death, Preprint, hal-02962550, 2020.

[33]

M. Mokhtar-Kharroubi and J. Banasiak, On spectral gaps of growth-fragmentation semigroups in higher moment spaces, Kinetic & Related Models, to appear.

[34]

R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Springer-Verlag Berlin, 1986. doi: 10.1007/BFb0074922.

[35]

B. Perthame, Transport Equations in Biology. Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007.

[36]

G. Schluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[37]

J. Voigt, On resolvent positive operators and positive $ C_{0}$-semigroups on AL-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.

[38]

G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.  doi: 10.2307/2000695.

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