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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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