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Asymptotic behavior of \(C_ 0\)-semigroups in B-convex spaces. (English) Zbl 0653.47018
The asymptotic behavior of \(C_ 0\)-semigroups in B-convex spaces is characterized in terms of growth conditions of the generator-resolvent. The main result improves a corresponding result of M. Slemrod [Indiana Univ. Math. J. 25, 783-792 (1976; Zbl 0313.47026)]. It is shown that the given characterization is optimal in B-convex spaces. Especially, Ljapunov’s stability condition reads as follows in B-convex spaces: Consider the abstract Cauchy problem \[ (ACP)\quad (d/dt- A)u=0,\quad u(0)\in D(A) \] associated with a \(C_ 0\)-semigroup \((U_ A(t))_{t\geq 0}\). If \(s_ b(A)<0\), \(s_ b(A)\) denoting the abscissa of boundedness of the resolvent \(z\mapsto (z-A)^{-1}\), then all classical solutions of (ACP) are uniformly exponentially decreasing. One should remark that for example all uniformly convex Banach spaces are B-convex, and therefore the main result holds in many classical spaces of functions and distributions.
Reviewer: V.Wrobel

MSC:
47D03 Groups and semigroups of linear operators
35F10 Initial value problems for linear first-order PDEs
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
35P05 General topics in linear spectral theory for PDEs
34G10 Linear differential equations in abstract spaces
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