Properties of the semigroup in \(L_1\) associated with age-structured diffusive properties. (English) Zbl 07785977
Summary: The linear semigroup associated with age-structured diffusive populations is investigated in the \(L_1\)-setting. A complete determination of its generator is given along with detailed spectral information that imply, in particular, an asynchronous exponential growth of the semigroup. Moreover, regularizing effects inherited from the diffusion part are exploited to derive additional properties of the semigroup.
MSC:
| 47D06 | One-parameter semigroups and linear evolution equations |
| 47A10 | Spectrum, resolvent |
| 35K90 | Abstract parabolic equations |
| 35M10 | PDEs of mixed type |
| 92D25 | Population dynamics (general) |
Keywords:
age structure; semigroups of linear operators; parabolic evolution operators; asynchronous exponential growthReferences:
| [1] | H. AMANN, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory, Mono-graphs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. https://dx.doi. org/10.1007/978-3-0348-9221-6. MR1345385 · Zbl 0819.35001 · doi:10.1007/978-3-0348-9221-6.MR1345385 |
| [2] | A. BÁTKAI, M. KRAMAR FIJAVŽ, and A. RHANDI, Positive Operator Semigroups: From Finite to Infinite Dimensions, with a foreword by R. NAGEL and U. SCHLOTTERBECK, Operator Theory: Advances and Applications, vol. 257, Birkhäuser/Springer, Cham, 2017. https://dx. doi.org/10.1007/978-3-319-42813-0. MR3616245 · Zbl 1420.47001 · doi:10.1007/978-3-319-42813-0.MR3616245 |
| [3] | Ph. CLÉMENT, H. J. A. M. HEIJMANS, S. ANGENENT, C. J. VAN DUIJN, and B. DE PAGTER, One-parameter Semigroups, CWI Monographs, vol. 5, North-Holland Publishing Co., Amsterdam, 1987. MR915552 · Zbl 0636.47051 · doi:10.1007/978-3-0348-9221-6 |
| [4] | D. DANERS and P. KOCH MEDINA, Abstract Evolution Equations, Periodic Problems and Appli-cations, Pitman Research Notes in Mathematics Series, vol. 279, Longman Scientific & Tech-nical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR1204883 · doi:10.1007/978-3-319-42813-0 |
| [5] | A. DUCROT and P. MAGAL, A center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the Gurtin-McCamy equation, Trans. Amer. Math. Soc. 372 (2019), no. 5, 3487-3537. https://dx.doi.org/10.1090/tran/ 7780. MR3988617 · Zbl 1422.37057 · doi:10.1090/tran/7780.MR3988617 |
| [6] | A. DUCROT, P. MAGAL, and A. THOREL, An integrated semigroup approach for age structured equations with diffusion and non-homogeneous boundary conditions, NoDEA Nonlinear Differen-tial Equations Appl. 28 (2021), no. 5, Paper No. 49, 31 pp. https://dx.doi.org/10.1007/ s00030-021-00710-x. MR4280529 · Zbl 1489.47066 · doi:10.1007/s00030-021-00710-x.MR4280529 |
| [7] | K.-J. ENGEL and R. NAGEL, One-parameter Semigroups for Linear Evolution Equations, with contributions by S. BRENDLE, M. CAMPITI, T HAHN, G. METAFUNE, G. NICKEL, D. PALLARA, C. PERAZZOLI, A. RHANDI, S. ROMANELLI, and R. SCHNAUBELT, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000. MR1721989 · doi:10.1090/tran/7780 |
| [8] | S. GUTMAN, Compact perturbations of m-accretive operators in general Banach spaces, SIAM J. Math. Anal. 13 (1982), no. 5, 789-800. https://dx.doi.org/10.1137/0513054. MR668321 · Zbl 0499.47046 · doi:10.1007/s00030-021-00710-x |
| [9] | M. GYLLENBERG and G. F. WEBB, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl. 167 (1992), no. 2, 443-467. https://dx.doi.org/10.1016/ 0022-247X(92)90218-3. MR1168600 · Zbl 0818.47064 |
| [10] | E. HILLE and R. S. PHILLIPS, Functional Analysis and Semi-groups, revised ed, American Math-ematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, RI, 1957. MR0089373 · Zbl 0078.10004 |
| [11] | H. KANG and S. RUAN, Mathematical analysis on an age-structured SIS epidemic model with nonlocal diffusion, J. Math. Biol. 83 (2021), no. 1, Paper No. 5, 30 pp. https://dx.doi.org/ 10.1007/s00285-021-01634-x. MR4278447 · doi:10.1016/0022-247X(92)90218-3 |
| [12] | , Nonlinear age-structured population models with nonlocal diffusion and nonlocal boundary conditions, J. Differential Equations 278 (2021), 430-462. https://dx.doi.org/10.1016/j. jde.2021.01.004. MR4203556 · Zbl 1456.35083 · doi:10.1016/j.jde.2021.01.004.MR4203556 |
| [13] | M. LANGLAIS, Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Biol. 26 (1988), no. 3, 319-346. https://dx.doi.org/10.1007/ BF00277394. MR949096 · Zbl 0713.92019 · doi:10.1007/s00285-021-01634-x |
| [14] | A. RHANDI, Positivity and stability for a population equation with diffusion on L 1 , Positivity 2 (1998), no. 2, 101-113. https://dx.doi.org/10.1023/A:1009721915101. MR1656866 · Zbl 0918.34058 · doi:10.1016/j.jde.2021.01.004 |
| [15] | A. RHANDI and R. SCHNAUBELT, Asymptotic behaviour of a non-autonomous population equa-tion with diffusion in L 1 , Discrete Contin. Dynam. Systems 5 (1999), no. 3, 663-683. https:// dx.doi.org/10.3934/dcds.1999.5.663. MR1696337 · Zbl 1002.92016 · doi:10.1016/j.jde.2021.01.004 |
| [16] | G. SIMONETT and C. WALKER, On the solvability of a mathematical model for prion proliferation, J. Math. Anal. Appl. 324 (2006), no. 1, 580-603. https://dx.doi.org/10.1016/j.jmaa. 2005.12.036. MR2262493 · Zbl 1109.92021 · doi:10.1007/BF00277394 |
| [17] | H. R. THIEME, Positive perturbation of operator semigroups: Growth bounds, essential compactness, and asynchronous exponential growth, Discrete Contin. Dynam. Systems 4 (1998), no. 4, 735-764. https://dx.doi.org/10.3934/dcds.1998.4.735. MR1641201 · Zbl 0988.47024 · doi:10.3934/dcds.1998.4.735.MR1641201 |
| [18] | Ch. WALKER, Global existence for an age and spatially structured haptotaxis model with nonlinear age-boundary conditions, European J. Appl. Math. 19 (2008), no. 2, 113-147. https://dx.doi. org/10.1017/S095679250800733X. MR2400718 · Zbl 1136.92023 · doi:10.1017/S095679250800733X.MR2400718 |
| [19] | , Age-dependent equations with non-linear diffusion, Discrete Contin. Dyn. Syst. 26 (2010), no. 2, 691-712. https://dx.doi.org/10.3934/dcds.2010.26.691. MR2556504 · Zbl 1184.35201 · doi:10.3934/dcds.2010.26.691.MR2556504 |
| [20] | , On positive solutions of some system of reaction-diffusion equations with nonlocal initial con-ditions, J. Reine Angew. Math. 660 (2011), 149-179. https://dx.doi.org/10.1515/crelle. 2011.074. MR2855823 · Zbl 1242.35141 · doi:10.3934/dcds.1998.4.735 |
| [21] | , Some remarks on the asymptotic behavior of the semigroup associated with age-structured diffusive populations, Monatsh. Math. 170 (2013), no. 3-4, 481-501. https://dx.doi.org/ 10.1007/s00605-012-0428-3. MR3055799 · doi:10.3934/dcds.2010.26.691 |
| [22] | , Some results based on maximal regularity regarding population models with age and spatial structure, J. Elliptic Parabol. Equ. 4 (2018), no. 1, 69-105. https://dx.doi.org/10.1007/ s41808-018-0010-9. MR3797184 · Zbl 1387.35373 · doi:10.1515/crelle.2011.074 |
| [23] | G. F. WEBB, Theory of Nonlinear Age-dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, vol. 89, Marcel Dekker, Inc., New York, 1985. MR772205 · doi:10.1007/s00605-012-0428-3 |
| [24] | , Population models structured by age, size, and spatial position, Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., vol. 1936, Springer, Berlin, 2008, pp. 1-49. https://dx.doi.org/10.1007/978-3-540-78273-5_1. MR2433574 · doi:10.1007/s41808-018-0010-9 |
| [25] | D-30167 Hannover Germany E-MAIL: walker@ifam.uni-hannover.de KEY WORDS AND PHRASES: Age structure, semigroups of linear operators, parabolic evolution operators, asynchronous exponential growth. 2010 MATHEMATICS SUBJECT CLASSIFICATION: 47D06, 47A10, 35K90, 35M10, 92D25. Received: September 7, 2021. · doi:10.1007/978-3-540-78273-5_1 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
