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
Abstract. In this paper, we characterize wellposedness of nonautonomous, linear Cauchy problemsa Banach space X by the existence of certain evolution semigroups.Then, we use these generation results for evolution semigroups to derive wellposedness for nonautonomous Cauchy problems under some "concrete" conditions. As a typical example, we discuss the so called "parabolic" case.
Basic definitionsIn this section, we introduce the basic definitions and notations in order to discuss nonautonomous Cauchy problems in terms of evolution families and evolution semigroups. In addition, we mention some of their fundamental properties.The solution of a nonautonomous Cauchy problem on some Banach space X can be given by a so called evolution family which can be defined as follows.
Definition 1.1 (Evolution family). A family (U (t, s)) t≥s of linear, bounded operators on a Banach space X is called an (exponentially bounded) evolution family iffor some M ≥ 1, ω ∈ IR and all t ≥ s ∈ IR.1991 Mathematics Subject Classification. Primary 47D06; secondary 47A55.
Abstract. The convergence of various operator splitting procedures, such as the sequential, the Strang and the weighted splitting, is investigated in the presence of a spatial approximation. To this end the relevant notions and results of numerical analysis are presented, a variant of Chernoff's product formula is proved and the general TrotterKato approximation theorem is used. The methods are applied to an abstract partial delay differential equation.
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