Long Time Behavior of Solution to Equal Mitosis PDE

Meas Len

doi:doi:10.11648/j.ijtam.20251105.11

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Long Time Behavior of Solution to Equal Mitosis PDE

Meas Len^*

Published in International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 5)

Received: 17 September 2025 Accepted: 9 October 2025 Published: 19 December 2025

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Abstract

In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 11, Issue 5)
DOI	10.11648/j.ijtam.20251105.11
Page(s)	71-77
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Equal Mitosis Equation, Floquet Theory, Entropy Method

References

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[3]	J. Clairambault, S. Gaubert and Th. Lepoutre, “Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models,” Math. Model. Nat. Phenom., vol 4(3), pp. 183-209, 2009.
[4]	T. Debiec, M. Doumic, P. Gwiazda and E. Wiedemann, “Relative entropy method for measure solutions of the growth-fragmentation equation,” SIAM J. Math. Anal., vol. 50(6), pp. 5811-5824, 2018.
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[6]	O. Diekmann, H.J.A.M. Heijmans and H. R. Thieme, “On the stability of cell size distribution II:Time periodic developmental rates,” Comp & Maths with Appls, vol. 12A, pp. 491-512, 1986.
[7]	M. Iannelli and F. Miler, “The basic approach to agestructured population dynamics,” Springer , pp. 51-52, 2017.
[8]	P. Laurenc¸ot and B. Perthame, “Exponential decay for the growth-fragmentation/cell-division equations,” Commun. Math. Sci, vol. 7(2), pp. 503-510, 2009.
[9]	Len Meas, “Spectral Analysis of Banach Spaces and their Application to Age-Structured Equations,” Cambodia Journal of Basic and Applied Research (CJBAR), vol. 3, pp. 32-69, 2021.
[10]	P. Gabriel and H. Martin, “Periodic asymptotic dynamics of the measure solutions to an equal mitosis equation,” Annales Henri Lebesgue, vol. 5, pp. 275-301, 2022.
[11]	P. Michel, “Existence of a solution to the cell division eigenproblem,” Math. Models Methods Appl. Sci., vol. 16, pp. 1125-1153, 2006.
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[13]	P. Michel, S. Mischler and B. Perthame, “General relative entropy inequality: an illustration on growth models,” J. Math. Pures Appl., vol. 84, pp.1235-1260, 2005.
[14]	S. Mischler and J. Scher, “Spectral analysis of semigroups and growth-fragmentation equations,” Annales de l’institut Henri Poincaré C, Analyse non linéaire, vol. 33(3), pp.849-898, 2016.
[15]	B. Perthame, “Transport equations in biology,” Frontiers in Mathematics, Birkh¨auser Verlag, Basel, 2007.
[16]	B. Perthame and L. Ryzhik, “Exponential decay for the fragmentation or cell division equation,” J. Differential Equations, vol. 210 (1), pp. 155-177, 2005.
[17]	G. F.Webb and A. Grabosch, “Asynchronous exponential growth in transition probability models of the cell cycle,” SIAM J. Math. Anal., vol. 18(4), pp. 897-908, 1987.

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APA Style

Len, M. (2025). Long Time Behavior of Solution to Equal Mitosis PDE. International Journal of Theoretical and Applied Mathematics, 11(5), 71-77. https://doi.org/10.11648/j.ijtam.20251105.11

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ACS Style

Len, M. Long Time Behavior of Solution to Equal Mitosis PDE. Int. J. Theor. Appl. Math. 2025, 11(5), 71-77. doi: 10.11648/j.ijtam.20251105.11

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AMA Style

Len M. Long Time Behavior of Solution to Equal Mitosis PDE. Int J Theor Appl Math. 2025;11(5):71-77. doi: 10.11648/j.ijtam.20251105.11

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@article{10.11648/j.ijtam.20251105.11,
  author = {Meas Len},
  title = {Long Time Behavior of Solution to Equal Mitosis PDE
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {5},
  pages = {71-77},
  doi = {10.11648/j.ijtam.20251105.11},
  url = {https://doi.org/10.11648/j.ijtam.20251105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251105.11},
  abstract = {In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.
},
 year = {2025}
}

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TY  - JOUR
T1  - Long Time Behavior of Solution to Equal Mitosis PDE

AU  - Meas Len
Y1  - 2025/12/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251105.11
DO  - 10.11648/j.ijtam.20251105.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 71
EP  - 77
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251105.11
AB  - In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.

VL  - 11
IS  - 5
ER  -

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Author Information

Meas Len

Department of Mathematics, Royal University of Phnom Penh, Phnom Penh, Cambodia

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http://orcid.org/0000-0001-9362-1575

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APA Style

Len, M. (2025). Long Time Behavior of Solution to Equal Mitosis PDE. International Journal of Theoretical and Applied Mathematics, 11(5), 71-77. https://doi.org/10.11648/j.ijtam.20251105.11

Copy | Download

ACS Style

Len, M. Long Time Behavior of Solution to Equal Mitosis PDE. Int. J. Theor. Appl. Math. 2025, 11(5), 71-77. doi: 10.11648/j.ijtam.20251105.11

Copy | Download

AMA Style

Len M. Long Time Behavior of Solution to Equal Mitosis PDE. Int J Theor Appl Math. 2025;11(5):71-77. doi: 10.11648/j.ijtam.20251105.11

Copy | Download

@article{10.11648/j.ijtam.20251105.11,
  author = {Meas Len},
  title = {Long Time Behavior of Solution to Equal Mitosis PDE
},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {11},
  number = {5},
  pages = {71-77},
  doi = {10.11648/j.ijtam.20251105.11},
  url = {https://doi.org/10.11648/j.ijtam.20251105.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20251105.11},
  abstract = {In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - Long Time Behavior of Solution to Equal Mitosis PDE

AU  - Meas Len
Y1  - 2025/12/19
PY  - 2025
N1  - https://doi.org/10.11648/j.ijtam.20251105.11
DO  - 10.11648/j.ijtam.20251105.11
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 71
EP  - 77
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20251105.11
AB  - In this work, we are interested in the long time behavior of a solution to equal mitosis partial differential equation with positive and periodic coefficients. First, we prove the existence and uniqueness of solution of Floquet eigenvalue and its adjoint eigenvalue problem to the equal mitosis equation by using the fixed point theorem in the suitable L1 weighted space under general division rate hypotheses. Let us recall that the Floquet exponent measures the growth rates of the population and understanding an eigenfunction is crucial for proving the long run behavior of the Cauchy problem. Then we apply the generalized relative entropy method to derive such long time asymptotic behavior of the population density.

VL  - 11
IS  - 5
ER  -

Copy | Download