No abstract

We investigate polynomial decay of classical solutions of linear evolution equations. For bounded strongly continuous semigroups on a Banach space this property is closely related to polynomial growth estimates of the resolvent of the generator. For systems of commuting normal operators polynomial decay is characterized in terms of the location of the generator spectrum. The results are applied to systems of coupled wave-type equations.

In this paper we convert a (linear abstract) initial boundary value problem into an abstract Cauchy problem on some product space and use semigroup methods to solve it. In particular, we apply spectral theory in order to discuss stability under boundary feedback. (2000). 47D06, 34G10, 93D15. Mathematics Subject Classification

We present a perturbation result for generators of C 0 -semigroups which can be considered as an operator theoretic version of the Weiss-Staffans perturbation theorem for abstract linear systems. The results are illustrated by applications to the Desch-Schappacher, the Miyadera-Voigt perturbation theorems, and to unbounded perturbations of the boundary conditions of a generator.

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in R m defined bygenerates an analytic semigroup of angle π 2 on C( ) for every β > 0 and γ ∈ C(∂ ) (for the definition of C 1 n ( ) cf. (1.3)). Introduction.Recently various authors have studied the operator A defined in (1) on spaces of continuous functions, see [3, Sect. 3], [4], [11]. While in these works only the generator property of A is shown, in this paper we prove that the generated semigroup is analytic. The proof is based on similarity transformations and perturbation arguments which were already used to treat second order differential operators on C[0, 1], see [8]. Our approach allows, in a certain sense, to completely decouple the interior from the boundary of . In this way we obtain two operators, the Dirichlet Laplacianon C( ) representing the dynamic in the interior and the Dirichlet-Neumann operator N (see Section 2) acting on the "boundary space" C(∂ ). Since A 0 and −N are both sectorial the generation result follows by a standard perturbation argument.