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Asymptotic behavior of  -semigroups
in Banach spaces
Authors:
Lutz Weis and Volker Wrobel
Journal:
Proc. Amer. Math. Soc. 124 (1996), 3663-3671
MSC (1991):
Primary 47D06
MathSciNet review:
1327051
Full-text PDF Free Access
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Additional Information
Abstract: We present optimal estimates for the asymptotic behavior of
strongly continuous semigroups  in terms of growth abscissas of the resolvent function  of the generator  . In particular we give Ljapunov's classical stability condition a definite form for (infinite dimensional) abstract Cauchy problems: The abscissa of boundedness of  equals the growth bound of the classical solutions of  .
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(1989), no. 1, 101–114. MR 982572
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- L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236 (1978), 385-394. MR 57:1191
- [Ko]
- H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285-346. MR 34:1862
- [vN-S-W]
- J. van Neerven, B. Straub, and L. Weis, On the asymptotic behaviour of a semigroup of linear operators, Indag. Math. 6 (1995), 453-476. CMP 96:05
- [Pa]
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer, Berlin, Heidelberg, and New York, 1983. MR 85g:47061
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- J. Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Math. Univ. Padova 42 (1969), 15-26. MR 41:812
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- M. Slemrod, Asymptotic behavior of
 -semigroups as determined by the spectrum of the generator, Indiana Univ. Math. J. 25 (1976), 783-892. MR 56:9321
- [Tr]
- H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam, New York, and Oxford, 1978. MR 80i:46032b
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- L. Weis, Stability of positive semigroups on
 , Proc. Amer. Math. Soc. 123 (1995), 3089-3094. MR 95m:47074
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- G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. Anal. Appl. 145 (1990), 154-171. MR 90k:47092
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- V. Wrobel, Asymptotic behavior of
 -semigroups in  -convex spaces, Indiana Univ. Math. J. 38 (1989), 101-113. MR 90b:47076
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Additional Information
Lutz Weis
Affiliation:
Mathematisches Institut I, Universität Karlsruhe, D-76128 Karlsruhe, Germany
Email:
lutz.weis@math.uni-karlsruhe.de
Volker Wrobel
Affiliation:
Mathematisches Seminar, Universität Kiel, D-24098 Kiel, Germany
DOI:
http://dx.doi.org/10.1090/S0002-9939-96-03373-4
PII:
S 0002-9939(96)03373-4
Received by editor(s):
January 17, 1995
Additional Notes:
The first author was supported by the Louisiana Education Quality Support Fund (LEQSF-RD-A-O8).
Communicated by:
Palle E. T. Jorgensen
Article copyright:
© Copyright 1996 American Mathematical Society
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