The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions

Räbiger, Frank; Schnaubelt, Roland

doi:10.1007/bf02574098

Cited by 63 publications

(30 citation statements)

References 14 publications

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“…With this set of seminorms one can see (see [13,Chapter 2,§20]) that L 1,loc (R + ) is a Fréchet space.…”

Section: Function Spaces Admissibility and Exponential Dichotomymentioning

confidence: 97%

Invariant manifolds of admissible classes for semi-linear evolution equations

Nguyen

2009

Journal of Differential Equations

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Consider an evolution family U = (U (t, s)) t s 0 on a half-line R + and a semi-linear integral equationWe prove the existence of invariant manifolds of this equation. These manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of L p type, the Lorentz spaces L p,q and many other function spaces occurring in interpolation theory. The existence of such manifolds is obtained in the case that (U (t, s)) t s 0 has an exponential dichotomy and the nonlinear forcing term f (t, x) satisfies the nonuniform Lipschitz conditions: f (t, x 1 ) − f (t, x 2 ) ϕ(t) x 1 − x 2 for ϕ being a real and positive function which belongs to certain classes of admissible function spaces.

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“…With this set of seminorms one can see (see [13,Chapter 2,§20]) that L 1,loc (R + ) is a Fréchet space.…”

Section: Function Spaces Admissibility and Exponential Dichotomymentioning

confidence: 97%

Invariant manifolds of admissible classes for semi-linear evolution equations

Nguyen

2009

Journal of Differential Equations

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“…These semigroups characterize the behavior of the evolution family completely and, consequently, will be called evolution semigroups. Evolution semigroups, first introduced in 1974 by Howland [5] and studied in 1976 by Evans [2], recently attracted a great deal of interest, see, e. g., [8], [9], [10], [13], [15], [16]. In particular, it was possible to characterize certain asymptotic behavior of evolution families by spectral properties of the corresponding evolution semigroup and its generator.…”

Section: Definition 11 (Evolution Family) a Family (U (T S)) T≥s Omentioning

confidence: 99%

Evolution semigroups for nonautonomous Cauchy problems

Nickel

1997

Abstract and Applied Analysis

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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.

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“…In this context, in the spirit of Perron's observations, the asymptotic properties of U like uniform exponential stability, uniform exponential dichotomy can be described as spectral properties of the infinitesimal generator of T (see [7,8,[18][19][20]25,26]). However, we have to remark that most results in this direction are restricted to the line case J = R. Also, the case of real half-line has been considered in [2,27].…”

Section: Introductionmentioning

confidence: 99%

“…More recently another approach uses frequently the so-called evolution semigroup T = {T (t)} t 0 on some suitable space of X-valued functions induced by the evolutionary process (see the papers due to N. van Minh [25,26], F. Räbiger and R. Schnaubelt [18,19], C. Chicone and Y. Latushkin [2], Y. Latushkin and T. Randolph [8], R. Rau [20], R. Schnaubelt [23], Y. Latushkin and S. Montgomery-Smith [7], Y. Latushkin et al [9], N. van Minh et al [27]). In fact, the construction of the evolution semigroup is surprisingly simple.…”

Section: Introductionmentioning

confidence: 99%

Schäffer spaces and exponential dichotomy for evolutionary processes

Preda

Pogan

Preda

2006

Journal of Differential Equations

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A very general characterization of exponential dichotomy for evolutionary processes in terms of the admissibility of some pair of spaces which are translation invariant (the so-called Schäffer spaces) is given. It includes, as particular cases, many interesting situations among which we note the results obtained by N. van Minh, F. Räbiger and R. Schnaubelt and the authors concerning the connections between admissibility and dichotomy.

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The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions

Cited by 63 publications

References 14 publications

Invariant manifolds of admissible classes for semi-linear evolution equations

Invariant manifolds of admissible classes for semi-linear evolution equations

Evolution semigroups for nonautonomous Cauchy problems

Schäffer spaces and exponential dichotomy for evolutionary processes

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