Abstract
The main purpose of this paper is to use ideas from systems theory to investigate the concept of maximal \(L^p\)-regularity for some perturbed autonomous and non-autonomous evolution equations in Banach spaces. We mainly consider two classes of perturbations: Miyadera–Voigt perturbations and Desch–Schappacher perturbations. We introduce conditions for which the maximal \(L^p\)-regularity can be preserved under these kinds of perturbations. We illustrate our results with some applications, in particular with an example of PDE in non-reflexive spaces.
Similar content being viewed by others
References
M. Adler, M. Bombieri, and K.-J. Engel. Perturbation of analytic semigroups and applications to partial differential equations, J. Evol. Equ. 17:1183–1208, 2017.
H. Amann. Linear and Quasilinear Parabolic Problems. Vol. I. Birkhäuser, Basel XX, 1995.
H. Amann, M. Hieber and G. Simonett. Bounded \(H^\infty \)-calculus for elliptic operators. Differ. Int. Equ., 7(3-4):613–653, 1994.
W. Arendt, R. Chill, S. Fornaro and C. Poupaud. \(L^p\)-maximal regularity for nonautonomous evolution equations. J. Differ. Equ., 237:1–26, 2007.
P. Cannarsa and V. Vespri. On maximal Lp regularity for the abstract Cauchy problem, Boll. Un. Mat. Ital. B, 5(6):165–175, 1986.
C. Chicone and Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations. American Mathematical Society., Providence, RI, 1999.
T. Coulhon and D. Lamberton, Régularité Lp pour les équations d’évolution, Séminaire d’Analyse Fonctionnelle 1984/1985, Publ. Math. Univ. Paris VII, 26:155–165, 1986.
L. De Simon, Un’ applicazione della theoria degli integrali singolariallo studio delle equazioni differenziali lineare astratte del primoordine. Rend. Sem. Mat., Univ. Padova, 99:205–223, 1964.
R. Denk, M. Hieber, and J. Prüss, R-Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type. Memoirs Am. Math. Soc., 166, Amer. Math. Soc., Providence, R.I., 2003.
G. Dore, Maximal regularity in Lp spaces for an abstract Cauchy problem. Adv. Differ. Equ. 5(1-3):293–322, 2000.
K.-J. Engel and R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000.
G. Greiner. Perturbing the boundary conditions of a generator. Houston J. Math., 18:405–425, 2001.
S. Hadd. An evolution equation approach to nonautonomous linear systems with state, input, and output delays. SIAM J Control Optim. 45:246–272, 2006
S. Hadd. Unbounded perturbations of \(C_0\)-semigroups on Banach spaces and applications. Semigroup Forum, 70:451–465, 2005.
S. Hadd and A. Idrissi, On the admissibility of observation for perturbed \(C_0\)–semigroups on Banach spaces, Syst. Control Lett. 55(1):1–7, 2006.
S. Hadd, R. Manzo and A. Rhandi. Unbounded perturbations of the generator domain. Discret. Cont. Dyn. Syst. A, 35(2):703–723, 2015.
B.H. Haak, M. Haase and P.C. Kunstmann. Perturbation, interpolation and maximal regularity. Adv. Differ. Equ., 11(2):201–240, 2006.
M. Hieber and S. Monniaux. Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations. Proc. Am. Math. Soc. 128:1047–1053, 2000.
P. C. Kunstmann and L. Weis. Maximal \(L_p\)-regularity for parabolic equations. Fourier multiplier theorems and \(H^\infty \) functional calculus. In finctional analytic methodes for evolution equations (Levico terme 2001), Lecture notes in Math., Springer, Berlin, 1855:65–312, 2004.
P. C. Kunstmann and L. Weis. Perturbation theorems for maximal Lp-regularity. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30(4):415–4435, 2001.
C. Le Merdy. The Weiss conjecture for bounded analytic semigroups. J. Lond. Math. Soc. 67(3):715–738, 2003.
F. Maragh, H. Bounit, A. Fadili and H. Hammouri. On the admissible control operators for linear and bilinear systems and the Favard spaces. Bull. Belgian Math. Soc. Simon Stevin, 21(4):711–732, 2014.
I. Miyadera, On perturbation theory for semigroups of operators. Tohoku Math. J. 18, pp. 299–310, 1966.
T. Ogawa and S. Shimizu. End-point maximal regularity and its application to two-dimensional Keller–Segel system. Math. Z. 264(3):601–628, 2010.
A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin, 1983.
J. Prüss and H. Sohr. Imaginary powers of elliptic second order differential operators in \(L^p\) -spaces. Hiroshima Math. J. 23(1):161–192, 1993.
P. Portal and Ž. Štrkalj. Pseudodifferential operators on Bochner spaces and an application. Math. Z. 253:805–819, 2006.
J. Prüss and R. Schnaubelt. Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time. J. Math. Anal. Appl. 256:405-430, 2001.
M. Renardy and R. C. Rogers, An introduction to partial differential equations, 2nd ed., Texts in Applied Mathematics, vol. 13, Springer-Verlag, New York, 2004.
D. Salamon. Infinite-dimensional linear system with unbounded control and observation: a functional analytic approach. Trans. Am. Math. Soc., 300(2):383–431, 1987.
R. Schnaubelt, Well-posedness and asymptotic behavior of non-autonomous linear evolution equations. Evolution Equations. Semigroups and Functional Analysis (A. Lorenzi and B. Ruf eds). Basel: Birkhäuser, 50:311–338, 2002.
R. Schnaubelt , Feedbacks for nonautonomous regular linear systems, SIAM J. Control Optim., 41(4):1141–1165, 2002.
O.J. Staffans, Well-Posed Linear Systems, vol. 103 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2005.
M. Tucsnak and G. Weiss. Observation and Control for Operator Semigroups, Birkhäuser, Basel, 2009.
J. Voigt. On the perturbation theory for strongly continuous semigroups. Math. Ann. 229(2):163–171, 1977.
L. Weis. A new approach to maximal \(L_p\)-regularity. Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 195–214, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.
L. Weis. Operator-valued Fourier multiplier theorems and maximal Lp-regularity, Math. Ann., 319:735–758, 2001.
G. Weiss. Admissible observation operators for linear semigroups, Israel J. Math. 65:17–43, 1989.
G. Weiss. Admissibility of unbounded control operators, SIAM J. Control Optim. 27:527–545, 1989.
G. Weiss. Regular linear systems with feedback. Mathematics of Control. Signals Syst., 7:23–57, 1994.
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.
Rights and permissions
About this article
Cite this article
Amansag, A., Bounit, H., Driouich, A. et al. On the maximal regularity for perturbed autonomous and non-autonomous evolution equations. J. Evol. Equ. 20, 165–190 (2020). https://doi.org/10.1007/s00028-019-00514-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-019-00514-8