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Asymptotic behavior of $$C_ 0$$-semigroups in Banach spaces. (English) Zbl 0863.47027
Summary: We present optimal estimates for the asymptotic behavior of strongly continuous semigroups $$U_A:[0,\infty[\to L(X)$$ in terms of growth abscissas of the resolvent function $$R(\cdot,A)$$ of the generator $$A$$. In particular, we give Lyapunov’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 for the classical solutions of $$y'=Ay$$.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations
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##### References:
 [1] Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385 – 394. · Zbl 0326.47038 [2] Hikosaburo Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285 – 346. · Zbl 0154.16104 [3] 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 · Zbl 0849.47018 [4] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. · Zbl 0516.47023 [5] Jaak Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (1969), 15 – 26 (French). · Zbl 0241.46033 [6] Marshall Slemrod, Asymptotic behavior of \?$$_{0}$$ semi-groups as determined by the spectrum of the generator, Indiana Univ. Math. J. 25 (1976), no. 8, 783 – 792. · Zbl 0313.47026 · doi:10.1512/iumj.1976.25.25062 · doi.org [7] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. [8] Lutz Weis, The stability of positive semigroups on \?_\? spaces, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3089 – 3094. · Zbl 0851.47028 [9] George Weiss, The resolvent growth assumption for semigroups on Hilbert spaces, J. Math. Anal. Appl. 145 (1990), no. 1, 154 – 171. · Zbl 0693.47034 · doi:10.1016/0022-247X(90)90438-L · doi.org [10] Volker Wrobel, Asymptotic behavior of \?$$_{0}$$-semigroups in \?-convex spaces, Indiana Univ. Math. J. 38 (1989), no. 1, 101 – 114. · Zbl 0653.47018 · doi:10.1512/iumj.1989.38.38004 · doi.org
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