Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.

Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.

The main result is the following stability theorem: Let T = ( T ( t ) ) t ⩾ 0 \mathcal {T} = {(T(t))_{t \geqslant 0}} be a bounded C 0 {C_0} -semigroup on a reflexive space X X . Denote by A A the generator of T \mathcal {T} and by σ ( A ) \sigma (A) the spectrum of A A . If σ ( A ) ∩ i R \sigma (A) \cap i{\mathbf {R}} is countable and no eigenvalue of A A lies on the imaginary axis, then lim t → ∞ T ( t ) x = 0 {\lim _{t \to \infty }}T(t)x = 0 for all x ∈ X x \in X .

Let S(t) be a bounded strongly continuous semi-group on a Banach space B and −A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1) −1 tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities.In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1) −1 , linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability). BackgroundConsider a strongly continuous semi-group S(t) on a Banach space B, with generator −A. Assume that S(t) = e −tA is bounded, i.e. sup t≥0 e −tA = C < ∞.(1.1) (Throughout the article, a semi-group will be strongly continuous on [0, ∞), i.e., a C 0 -semigroup. Moreover, · will denote both the norm on B and the operator norm from B to B.) The operator A is closed and densely defined, and we denote by D(A) its domain, σ(A) its spectrum and ρ(A) its resolvent set. It is a well-known property that if (1.1) holds, then the left open half-plane {Re z < 0} is included in ρ(A) (see [25,11]). In 1988, Lyubich and Vũ [20] and Arendt and Batty [1] have shown that if σ(A) ∩ iR is countable and σ(A * ) ∩ iR contains no eigenvalue (here A * is the adjoint of A), then the semi-group is (pointwise) strongly stable, that ist→+∞ e −tA u 0 = 0. For surveys of this and other results concerning strong stability, see [4], [8] 1 .This work was partially supported by the French ANR ControlFlux. The second author would like to thank Nicolas Burq for fruitful discussions on the subject, and Luc Miller for pointing out the stability theorem of Lyubich, Vũ, Arendt and Batty and the article [3]. 1 We will only address strong stability concepts, and refer to [10] for a recent survey on different types of weak stability.

Vector-valued Laplace transforms and Cauchy problems 1 Wolfgang Arendt ... [et al.]. p. cm. -(Monographs in mathematics ; voi. 96) lncludes bibliographical references and index.