Abstract
This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille–Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a class of age-structured population models in \( L^p \) spaces (\( 1 \le p < \infty \)).
Similar content being viewed by others
Change history
24 May 2020
The original publication of the article contains errors which need to be amended as mentioned below.
References
W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd ed., Monographs in Mathematics, vol. 96, Birkhäuser/Springer Basel AG, Basel, 2011.
F. Andreu, J. Martínez, and J. M. Mazón, A spectral mapping theorem for perturbed strongly continuous semigroups, Math. Ann. 291 (1991), no. 3, 453–462.
W. Arendt, Resolvent positive operators, Proc. London Math. Soc. (3) 54 (1987), no. 2, 321–349.
S. Boulite, S. Hadd, and L. Maniar, Critical spectrum and stability for population equations with diffusion in unbounded domains, Discrete Contin. Dyn. Syst. Ser. B 5 (2005), no. 2, 265–276.
C. J. K. Batty and S. Król, Perturbations of generators of \(C_0\)-semigroups and resolvent decay, J. Math. Anal. Appl. 367 (2010), no. 2, 434–443.
M. D. Blake, A spectral bound for asymptotically norm-continuous semigroups, J. Operator Theory 45 (2001), no. 1, 111–130.
A. Bátkai, L. Maniar, and A. Rhandi, Regularity properties of perturbed Hille–Yosida operators and retarded differential equations, Semigroup Forum 64 (2002), no. 1, 55–70.
S. Brendle, R. Nagel, and J. Poland, On the spectral mapping theorem for perturbed strongly continuous semigroups, Arch. Math. (Basel) 74 (2000), no. 5, 365–378.
A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl. 264 (2001), no. 1, 1–20.
A. Benedek and R. Panzone, The space \(L^{p}\), with mixed norm, Duke Math. J. 28 (1961), 301–324.
S. Brendle, On the asymptotic behavior of perturbed strongly continuous semigroups, Math. Nachr. 226 (2001), 35–47.
A. N. Carvalho, T. Dlotko, and M. J. D. Nascimento, Non-autonomous semilinear evolution equations with almost sectorial operators, J. Evol. Equ. 8 (2008), no. 4, 631–659.
D. Chen, A linear theory for a class of delay equations with non-dense domains, 2018, in preparation.
D. Chen, The exponential dichotomy and invariant manifolds for some classes of differential equations, arXiv e-prints (Mar. 2019), available at arXiv:1903.08040
C. Chicone and Y. Latushkin, Evolution semigroups in dynamical systems and differential equations, Mathematical Surveys and Monographs, vol. 70, American Mathematical Society, Providence, RI, 1999.
M. G. Crandall and A. Pazy, On the differentiability of weak solutions of a differential equation in Banach space, J. Math. Mech. 18 (1968/1969), 1007–1016.
R. F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics, vol. 21, Springer-Verlag, New York, 1995.
K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985.
A. Ducrot, Z. Liu, and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl. 341 (2008), no. 1, 501–518.
A. Ducrot, P. Magal, and K. Prevost, Integrated semigroups and parabolic equations. Part I: linear perturbation of almost sectorial operators, J. Evol. Equ. 10 (2010), no. 2, 263–291.
G. Da Prato and E. Sinestrari, Differential operators with nondense domain, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), no. 2, 285–344 (1988).
K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000, With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.
I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, vol. 49, Birkhäuser Verlag, Basel, 1990.
P. S. Iley, Perturbations of differentiable semigroups, J. Evol. Equ. 7 (2007), no. 4, 765–781.
H. Kellerman and M. Hieber, Integrated semigroups, J. Funct. Anal. 84 (1989), no. 1, 160–180.
C. Kaiser and L. Weis, Perturbation theorems for \(\alpha \)-times integrated semigroups, Arch. Math. (Basel) 81 (2003), no. 2, 215–228.
T. Mátrai, On perturbations preserving the immediate norm continuity of semigroups, J. Math. Anal. Appl. 341 (2008), no. 2, 961–974.
G. A. Monteiro, On functions of bounded semivariation, Real Anal. Exchange 40 (2014/15), no. 2, 233–276.
P. Magal and S. Ruan, On integrated semigroups and age structured models in \(L^p\) spaces, Differential Integral Equations 20 (2007), no. 2, 197–239.
P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc. 202 (2009), no. 951, vi+71.
P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations 14 (2009), no. 11-12, 1041–1084.
P. Magal and S. Ruan, Theory and applications of abstract semilinear Cauchy problems, Applied Mathematical Sciences, vol. 201, Springer, Cham, 2018, With a foreword by Glenn Webb.
S. Mischler and J. Scher, Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 3, 849–898.
R. Nagel and J. Poland, The critical spectrum of a strongly continuous semigroup, Adv. Math. 152 (2000), no. 1, 120–133.
R. Nagel and S. Piazzera, On the regularity properties of perturbed semigroups, Rend. Circ. Mat. Palermo (2) Suppl. (1998), no. 56, 99–110, International Workshop on Operator Theory (Cefalù, 1997).
A. Pazy, On the differentiability and compactness of semigroups of linear operators, J. Math. Mech. 17 (1968), 1131–1141.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
S. Piazzera, Qualitative properties of perturbed semigroups, Ph.D. thesis, Verlag nicht ermittelbar, 1999.
F. Periago and B. Straub, A functional calculus for almost sectorial operators and applications to abstract evolution equations, J. Evol. Equ. 2 (2002), no. 1, 41–68.
M. Renardy, On the stability of differentiability of semigroups, Semigroup Forum 51 (1995), no. 3, 343–346.
A. Rhandi, Positivity and stability for a population equation with diffusion on \(L^1\), Positivity 2 (1998), no. 2, 101–113.
F. Räbiger, R. Schnaubelt, A. Rhandi, and J. Voigt, Non-autonomous Miyadera perturbations, Differential Integral Equations 13 (2000), no. 1-3, 341–368.
M. Sbihi, A resolvent approach to the stability of essential and critical spectra of perturbed \(C_0\)-semigroups on Hilbert spaces with applications to transport theory, J. Evol. Equ. 7 (2007), no. 1, 35–58.
R. Schnaubelt, Well-posedness and asymptotic behaviour of non-autonomous linear evolution equations, Evolution equations, semigroups and functional analysis (Milano, 2000), 2002, pp. 311–338.
R. Schnaubelt, Exponential bounds and hyperbolicity of evolution families, Ph.D. thesis, 1996.
H. R. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ. 8 (2008), no. 2, 283–305.
H. R. Thieme, Analysis of age-structured population models with an additional structure, Mathematical population dynamics (New Brunswick, NJ, 1989), 1991, pp. 115–126.
H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in mathematical population dynamics–molecules, cells and man (Houston, TX, 1995), 1997, pp. 691–711.
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.
H. R. Thieme and J. Voigt, Relatively bounded extensions of generator perturbations, Rocky Mountain J. Math. 39 (2009), no. 3, 947–969.
J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math. 90 (1980), no. 2, 153–161.
G. F. Webb, Population models structured by age, size, and spatial position, Structured population models in biology and epidemiology, 2008, pp. 1–49.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Part of this work was done at East China Normal University. The author would like to thank Shigui Ruan, Ping Bi and Dongmei Xiao for their useful discussions and encouragement. The author is grateful to the referee(s) for useful comments and suggestions and particularly pointing out Lemma 6.9 and a mistake in Theorem 7.4, which improved significantly the presentation of the original manuscript.
Rights and permissions
About this article
Cite this article
Chen, D. Regularity properties of some perturbations of non-densely defined operators with applications. J. Evol. Equ. 20, 659–702 (2020). https://doi.org/10.1007/s00028-019-00510-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-019-00510-y
Keywords
- Regularity
- Perturbation
- Non-densely defined operators
- Critical growth bound
- Essential growth bound
- Integrated semigroup
- Age-structured population model