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

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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

Mustapha Mokhtar-Kharroubi ^,

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 December 2020 Early access 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.

Keywords: Structured populations, Wentzell boundary condition, unbounded reproduction operator, weak compactness, asynchronous exponential growth.

Mathematics Subject Classification: Primary: 47D06, 92D25, 35B40, 35B50.

Citation: Mustapha Mokhtar-Kharroubi. Spectra of structured diffusive population equations with generalized Wentzell-Robin boundary conditions and related topics. Discrete and 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.
[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.
[3]	W. Desch, Perturbations of positive semigroups in AL-spaces, unpublished manuscript, 1988.
[4]	N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, John Wiley & Sons, 1988.
[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.
[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.
[7]	I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, Siam J Appl Math, 19 (1970), 607-628. doi: 10.1137/0119060.
[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.
[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.
[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.
[11]	R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, 1986.
[12]	B. de. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149–153. doi: 10.1007/BF01162028.
[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.
[14]	G. Scluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266. doi: 10.1007/BF01444620.
[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.
[16]	J. Voigt, On the convex compactness property for the strong operator topology, Note.di. Mat., 12 (1992), 259-269.
[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.

show all references

References:

[1]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011.
[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.
[3]	W. Desch, Perturbations of positive semigroups in AL-spaces, unpublished manuscript, 1988.
[4]	N. Dunford and J. T. Schwartz, Linear Operators, Part 1: General Theory, John Wiley & Sons, 1988.
[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.
[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.
[7]	I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, Siam J Appl Math, 19 (1970), 607-628. doi: 10.1137/0119060.
[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.
[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.
[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.
[11]	R. Nagel (Ed), One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184, Springer, 1986.
[12]	B. de. Pagter, Irreducible compact operators, Math. Z, 192 (1986), 149–153. doi: 10.1007/BF01162028.
[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.
[14]	G. Scluchtermann, On weakly compact operators, Math. Ann., 292 (1992), 263-266. doi: 10.1007/BF01444620.
[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.
[16]	J. Voigt, On the convex compactness property for the strong operator topology, Note.di. Mat., 12 (1992), 259-269.
[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.

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