Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Asymptotic behavior of $C_0$-semigroups
in Banach spaces


Authors: Lutz Weis and Volker Wrobel
Journal: Proc. Amer. Math. Soc. 124 (1996), 3663-3671
MSC (1991): Primary 47D06
DOI: https://doi.org/10.1090/S0002-9939-96-03373-4
MathSciNet review: 1327051
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present optimal estimates for the asymptotic behavior of
strongly continuous semigroups $U_A:[0,\infty [\rightarrow L(X)$ in terms of growth abscissas of the resolvent function $R(\cdot ,A)$ of the generator $A$. In particular we give Ljapunov's classical stability condition a definite form for (infinite dimensional) abstract Cauchy problems: The abscissa of boundedness of $R(\cdot ,A)$ equals the growth bound of the classical solutions of $y'=Ay$.


References [Enhancements On Off] (What's this?)

  • [G] 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
  • [P] J. Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Math. Univ. Padova 42 (1969), 15-26. MR 41:812
  • [Sl] M. Slemrod, Asymptotic behavior of $C_0$-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
  • [Ws] L. Weis, Stability of positive semigroups on $L_p(\mu )$, Proc. Amer. Math. Soc. 123 (1995), 3089-3094. MR 95m:47074
  • [Wss] G. Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. Anal. Appl. 145 (1990), 154-171. MR 90k:47092
  • [Wr] V. Wrobel, Asymptotic behavior of $C_0$-semigroups in $B$-convex spaces, Indiana Univ. Math. J. 38 (1989), 101-113. MR 90b:47076

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D06

Retrieve articles in all journals with MSC (1991): 47D06


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: https://doi.org/10.1090/S0002-9939-96-03373-4
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

American Mathematical Society