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A condition of uniform exponential stability for semigroups. (English) Zbl 1180.47027
In the paper under review, relations between exponential stability of semigroups and their weak stability are investigated. It has been proven before that weak-\(L^p\)-stability of a semigroup \(T\) in a Hilbert space \(H\) is equivalent to its exponential stability. (\(T\) is called weakly-\(L^p\)-stable if \( \int_0^\infty|\langle T(t)x, y\rangle|^p\, dt<\infty\) for each \(x,y\in H,\) and \(T\) is uniformly exponentially stable if
\[ \omega_0(T) := \lim_{t\to\infty}{{\log\|T(t)||}\over{t}} <0.) \] The author shows that a strongly continuous semigroup \(T\) in an Orlicz space \(E=(L^\Phi, \rho^\Phi)\) under certain conditions on the conjugate space \(E^*\) is uniformly exponentially stable if and only if it verifies the following estimate
\[ \sup_{\|x\|\leq1}\rho^\Phi(|\langle T(\cdot)x, x\rangle|)<\infty, \quad \text{where} \quad \rho^\Phi(f):=\inf\{k>0: \int_0^\infty \Phi (k^{-1}|f(t)|) \,dt\leq 1\}. \]

47D06 One-parameter semigroups and linear evolution equations
35B35 Stability in context of PDEs