Abstract
We first summarize some well-known, however instructive facts from the theory of autonomous abstract Cauchy problems for a closed operator (A,D(A)) on some Banach space X (compare [5], Chapter II.6).
To the Memory of Brunello Terreni
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander: Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser Verlag, 2001.
P. Acquistapace, B. Terreni: A unified approach to abstract linear nonau-tonomous parabolic equations. Rend. Sem. Univ. Padova 78 (1987), 47–107.
C. Chicone, Y. Latushkin: Evolution Semigroups in Dynamical Systems and Differential Equations. Amer. Math. Soc, 1999.
G. Daprato, P. Grisvard: Sommes d’opérateurs linéaires et équations différentielles Operationelles. J. math, pures appl. 54 (1975), 305–387.
K.-J. Engel, R. Nagel: One-Parameter Semigroups for Linear Evolution Equations. Springer-Verlag, 2000.
D.E.E Vans: Time dependent perturbations and scattering of strongly continuous groups on Banach space. Math. Ann. 221 (1976), 275–290.
H. O. Fattorini: “The Cauchy Problem.” Addison Wesley, 1983.
J. Goldstein: Abstract evolution equations. Trans. Amer. Math. Soc. 141 (1969), 159–184.
J. Goldstein: On the absence of necessary conditions f or linear evolution operators. Amer. Math. Soc. 64 (1977), 77–80.
J. Hahn: “Nichtautonome hyperbolische Cauchyprobleme.” Diplomarbeit, Tübingen 1995.
J.S. Howland: Stationary scattering theory for time-dependent Hamiltonians. Math. Ann. 207 (1974), 315–335.
H. Kellermann: Linear evolution equations with time-dependent domain. Semes-terbericht Funktionalanalysis, 1985.
Y. Latushkin, S. Montgomery-Smith, and T. Randolph: Evolutionary semigroups and dichotomies of linear skew-product flows on locally compact spaces with Banach fibers. J. Diff. Equations 125 (1996), 73–116.
Y. Latushkin, T. W. Randolph: Dichotomy of differential equations on Banach spaces and an algebra of weighted translation operators. Integr. Eqns. Oper. Th. 23 (1995), 472–500.
S Moniaux, J. Prüss: A theorem of the Dore-Venni type for non-commuting operators. Tran. Amer. Math. Soc. 349 (1997), 4787–4814.
R. Nagel, G. Nickel, S. Romanelli: Identification of extrapolation spaces for unbounded operators. Quaestiones Math. 19 (1996), 83–100.
R. Nagel, A. Rhandi: A characterization of Lipschitz continuous evolution families on Banach spaces. Operator Theory: Advances and Application 75, Birkhäuser Verlag, Basel 1995, 275–288.
G. Nickel: “On Evolution Semigroups and Wellposedness of Nonautonomous Cauchy Problems.” PhD Thesis, Tübingen, 1996.
G. Nickel: Evolution semigroups for nonautonomous Cauchy problems. Abstract Appl. Analysis 2 (1997), 73–95.
G. Nickel, R. Schnaubelt: An extension of Kato’s stability condition for nonautonomous Cauchy problems. Taiwanese J. Math. 2 (1998), 483–496.
F. Räbiger, A. Rhandi, R. Schnaubelt, J. Voigt: Non-autonomous Miyadera perturbations. Differential Integral Equations 13 (1999), 341–368.
F. Räbiger, R. Schnaubelt: The spectral mapping theorem for evolution semigroups on spaces of vector valued functions. Semigroup Forum 52 (1996), 225–239.
R. Schnaubelt: Well-posedness and asymptotic behaviour of non-autonomous evolution equations. 2001.
P. E. Sobolevskii: Equations of parabolic type in a Banach space. Amer. Math. Soc. Transi. 49 (1966), 1–62.
H. Tanabe: Remarks on the equations of evolution in a Banach space. Osaka Math. J. 112 (1960), 145–166.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this chapter
Cite this chapter
Nagel, R., Nickel, G. (2002). Well-posedness for Nonautonomous Abstract Cauchy Problems. In: Lorenzi, A., Ruf, B. (eds) Evolution Equations, Semigroups and Functional Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 50. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8221-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8221-7_15
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9480-7
Online ISBN: 978-3-0348-8221-7
eBook Packages: Springer Book Archive