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December  2020, 13(12): 3551-3563. doi: 10.3934/dcdss.2020244

## Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics

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

* Corresponding author: Mustapha Mokhtar-Kharroubi

Received  January 2019 Published  January 2020

This paper provides two different extensions of a previous joint work "Time asymptotics of structured populations with diffusion and dynamic boundary conditions; Discrete Cont Dyn Syst, Series B, 23 (10) (2018)" devoted to asynchronous exponential asymptotics for bounded and weakly compact reproduction operators. The first extension considers bounded non weakly compact reproduction operators while the second extension deals with unbounded kernel reproduction operators and needs, as a preliminary step, a new generation result.

Citation: Mustapha Mokhtar-Kharroubi. Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3551-3563. doi: 10.3934/dcdss.2020244
##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar [2] P. Clement, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter., One-Parameter Semigroups, Vol. 5, North-Holland Publ Co., Amsterdam, 1987.  Google Scholar [3] W. Desch, Perturbations of positive semigroups in AL-spaces, unpublished manuscript, 1988. Google Scholar [4] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, John Wiley & Sons, 1988. Google Scholar [5] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Mathematical Biosciences and Engineering, 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.  Google Scholar [6] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.  Google Scholar [7] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, Siam J Appl Math, 19 (1970), 607-628.  doi: 10.1137/0119060.  Google Scholar [8] 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.  Google Scholar [9] M. Mokhtar-Kharroubi, On Schrödinger semigroups and related topics, J. Funct. Anal, 256 (2009), 1998-2025.  doi: 10.1016/j.jfa.2008.11.012.  Google Scholar [10] M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete Cont Dyn Syst, Series B, 23 (2018), 4087-4116.   Google Scholar [11] R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, 1986.  Google Scholar [12] B. de. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149–153. doi: 10.1007/BF01162028.  Google Scholar [13] A. Rhandi, Dyson-Phillips expansion and unbounded perturbations of linear $C_{0}$-semigroups, J. Computational Appl Math, 44 (1992), 339-349.  doi: 10.1016/0377-0427(92)90005-I.  Google Scholar [14] G. Scluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.  Google Scholar [15] J. Voigt, On resolvent positive operators and positive $C_{0}$-semigroups in AL-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.  Google Scholar [16] J. Voigt, On the convex compactness property for the strong operator topology, Note.di. Mat., 12 (1992), 259-269.   Google Scholar [17] L. Weis, A short proof for the stability theorem for positive semigroups on $Lp(\mu)$, Proceedings of the Amer Math Soc, 126 (1998), 3253-3256.  doi: 10.1090/S0002-9939-98-04612-7.  Google Scholar

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##### References:
 [1] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.  Google Scholar [2] P. Clement, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter., One-Parameter Semigroups, Vol. 5, North-Holland Publ Co., Amsterdam, 1987.  Google Scholar [3] W. Desch, Perturbations of positive semigroups in AL-spaces, unpublished manuscript, 1988. Google Scholar [4] N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, John Wiley & Sons, 1988. Google Scholar [5] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Mathematical Biosciences and Engineering, 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.  Google Scholar [6] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6 (1958), 261-322.  doi: 10.1007/BF02790238.  Google Scholar [7] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, Siam J Appl Math, 19 (1970), 607-628.  doi: 10.1137/0119060.  Google Scholar [8] 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.  Google Scholar [9] M. Mokhtar-Kharroubi, On Schrödinger semigroups and related topics, J. Funct. Anal, 256 (2009), 1998-2025.  doi: 10.1016/j.jfa.2008.11.012.  Google Scholar [10] M. Mokhtar-Kharroubi and Q. Richard, Time asymptotics of structured populations with diffusion and dynamic boundary conditions, Discrete Cont Dyn Syst, Series B, 23 (2018), 4087-4116.   Google Scholar [11] R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, 1986.  Google Scholar [12] B. de. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149–153. doi: 10.1007/BF01162028.  Google Scholar [13] A. Rhandi, Dyson-Phillips expansion and unbounded perturbations of linear $C_{0}$-semigroups, J. Computational Appl Math, 44 (1992), 339-349.  doi: 10.1016/0377-0427(92)90005-I.  Google Scholar [14] G. Scluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266.  doi: 10.1007/BF01444620.  Google Scholar [15] J. Voigt, On resolvent positive operators and positive $C_{0}$-semigroups in AL-spaces, Semigroup Forum, 38 (1989), 263-266.  doi: 10.1007/BF02573236.  Google Scholar [16] J. Voigt, On the convex compactness property for the strong operator topology, Note.di. Mat., 12 (1992), 259-269.   Google Scholar [17] L. Weis, A short proof for the stability theorem for positive semigroups on $Lp(\mu)$, Proceedings of the Amer Math Soc, 126 (1998), 3253-3256.  doi: 10.1090/S0002-9939-98-04612-7.  Google Scholar
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