非光滑边界条件下具时滞的Rotenberg方程主算子的谱分析

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1320-1331.

非光滑边界条件下具时滞的Rotenberg方程主算子的谱分析

童雅阁(),吴开谡*()

^{北京化工大学数理学院北京 100029}

收稿日期:2021-08-12 出版日期:2022-10-26 发布日期:2022-09-30
通讯作者: 吴开谡 E-mail:865066107@qq.com;wuks@mail.buct.edu.cn
作者简介:童雅阁, E-mail: 865066107@qq.com
基金资助:
北京化工大学精品课程项目(00810220)

Spectral Analysis of the Main Operator of Rotenberg Equation with Time Delay Under Nonsmooth Boundary Conditions

Yage Tong(),Kaisu Wu*()

^{School of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029}

Received:2021-08-12 Online:2022-10-26 Published:2022-09-30
Contact: Kaisu Wu E-mail:865066107@qq.com;wuks@mail.buct.edu.cn
Supported by:
the Excellent Course Project of Beijing University of Chemical Technology(00810220)

摘要/Abstract

摘要：

该文考虑细胞分裂的时滞性, 引入了具时滞的Rotenberg迁移方程, 在$L_1$空间中非光滑边界条件下(边界算子无界), 证明迁移算子生成$C_0$ -半群, 并进一步分析迁移算子的谱, 得到该迁移算子的谱在区域$\Gamma=\sigma(A_H)\cap \{\lambda\in C | {\rm Re}\lambda>\gamma\}$ ($\gamma>{\rm max}\{\lambda_0, -\sigma_0\}$)中仅由有限个具有有限代数重数的离散本征值组成等结果($\lambda_0, -\sigma_0$的具体意义见正文).

关键词: Rotenberg方程, 迁移算子, 非光滑边界条件, 谱分析, 时滞性

Abstract:

After considering the time delay of cell division, the Rotenberg transport equation with time-delay is introduced in this paper. In $L_1$ space, under the condition of non-smooth boundary (the boundary operator is unbounded), it is proved that the transport operator can generate a $C_0$-semigroup. Furthermore, we study the spectrum of the transport operator and prove that the region $\Gamma=\sigma(A_H)\cap \{\lambda\in C | {\rm Re}\lambda>\gamma\}$ ($\gamma>{\rm max}\{\lambda_0, -\sigma_0\}$) is only consists of a finite number of discrete eigenvalues with finite algebraic multiplicity.

Key words: Rotenberg equation, Transport operators, Nonsmooth boundary conditions, Spectrum analysis, Time delay

中图分类号:

O177.2

童雅阁,吴开谡. 非光滑边界条件下具时滞的Rotenberg方程主算子的谱分析[J]. 数学物理学报, 2022, 42(5): 1320-1331.

Yage Tong,Kaisu Wu. Spectral Analysis of the Main Operator of Rotenberg Equation with Time Delay Under Nonsmooth Boundary Conditions[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1320-1331.

参考文献 18

1	Rotenberg M . Transport theory for growing cell populations. Journal of Theoretical Biology, 1983, 103, 181- 199 doi: 10.1016/0022-5193(83)90024-3
2	吴红星, 王胜华, 袁邓彬. 种群细胞增生中迁移算子的谱研究. 系统科学与数学, 2017, 37 (3): 940- 949
	Wu H X , Wang S H , Yuan D B . The spectral research of transport operators arising in growing cell populations. Journal of Systems Science and Mathematical Sciences, 2017, 37 (3): 940- 949
3	Jeribi A , Megdiche H , Moalla N . On a transport operator arising in growing cell populations Ⅱ: Cauchy problem. Mathematical Methods in the Applied Sciences, 2005, 266, 70- 99
4	Boulanouar M . A mathematical study for a rotenberg model. Journal of Mathematical Analysis and Applications, 2002, 265, 371- 394 doi: 10.1006/jmaa.2001.7721
5	Lebowitz J L , Rubinow S I . A theory for the age and generation time distribution of a microbial population. Journal of Mathematical Biology, 1974, 1, 17- 36 doi: 10.1007/BF02339486
6	Piazzera S , Tonetto L . Asynchronous exponential growth for an age dependent population equation with delayed birth process. Journal of Evolution Equations, 2005, 5, 61- 77 doi: 10.1007/s00028-004-0159-6
7	Bai M , Xu S H . On a two-phase size-structured population model with infinite states-at-birth and distributed delay in birth process. Journal of Biological Dynamics, 2014, 8, 42- 56 doi: 10.1080/17513758.2014.899637
8	王胜华, 程国飞. 具结构化的细菌种群模型解的渐近行为. 数学物理学报, 2018, 38A (1): 156- 167
	Wang S H , Cheng G F . The asymptotic behavior of the solution in structured bacterial population model. Acta Mathematica Scientia, 2018, 38A (1): 156- 167
9	王胜华, 程国飞. 一类增生扩散型种群细胞中迁移方程的谱问题. 数学物理学报, 2013, 30A (1): 71- 77 doi: 10.3969/j.issn.1003-3998.2013.01.009
	Wang S H , Cheng G F . Spectral problem of transport equations with a proliferating cell population. Acta Mathematica Scientia, 2013, 30A (1): 71- 77 doi: 10.3969/j.issn.1003-3998.2013.01.009
10	Lods B , Mokhtar-Kharroubi M . On the theory of a growing cell population with zero minimum cycle length. Journal of Mathematical Analysis and Applications, 2002, 266, 70- 99 doi: 10.1006/jmaa.2001.7712
11	王胜华, 吴红星. Rotenberg模型中一类迁移算子的谱分析. 数学物理学报, 2016, 36A (5): 821- 831 doi: 10.3969/j.issn.1003-3998.2016.05.002
	Wang S H , Wu H X . Spectral analysis of a transport operators in rotenberg model. Acta Mathematica Scientia, 2016, 36A (5): 821- 831 doi: 10.3969/j.issn.1003-3998.2016.05.002
12	Latrach K . Compactness properties for perturbed semigroups and application to transport equation. Journal of the Australian Mathematical Society, 2000, 69 (1): 25- 40 doi: 10.1017/S1446788700001828
13	Latrach K , Mokhtar-Kharroubi M . On an unbounded linear operator arising in the theory of growing cell population. Journal of Mathematical Analysis and Applications, 1997, 211 (1): 273- 294 doi: 10.1006/jmaa.1997.5460
14	Van Neerven J M A M , Straub B , Weis L . On the asymptotic behaviour of a semigroup of linear operators. Indagationes Mathematicae, 1995, 6 (4): 453- 476 doi: 10.1016/0019-3577(96)81760-5
15	Rudnicki R . Asymptotic behaviour of a transport equation. Annales Polonici Mathematici, 1992, 57 (1): 45- 55
16	Dehici A , Jeribi A , Latrach K . Spectral analysis of a transport operator arising in growing cell populations. Acta Applicandae Mathematicae, 2006, 92, 37- 62
17	王胜华, 翁云芳, 阳名珠. 人体细胞增生中一类迁移算子的谱分析. 数学物理学报, 2010, 30A (4): 1055- 1061
	Wang S H , Weng Y F , Yang M Z . Spectral problem of transport equations with a proliferating cell population. Acta Mathematica Scientia, 2010, 30A (4): 1055- 1061
18	阳名珠. 线性迁移理论中各类本征值问题的研究. 山西大学学报, 1983, 4, 92- 106
	Yang M Z . Study of vavious eigenvalue problems in linear transport theory. Journal of Shanxi University, 1983, 4, 92- 106

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