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Asymptotic behaviour of parabolic nonautonomous evolution equations, Functional analytic methods for evolution equations
 Lecture Notes in Math
, 2004
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Integral characterizations for exponential stability of semigroups and evolution families on Banach spaces
, 2006
"... Abstract. Let X be a real or complex Banach space and U = {U(t, s)}t≥s≥0 be a strongly continuous and exponentially bounded evolution family on X. Let J be a nonnegative functional on the positive cone of the space of all realvalued locally bounded functions on R+: = [0,∞). We suppose that J satis ..."
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Abstract. Let X be a real or complex Banach space and U = {U(t, s)}t≥s≥0 be a strongly continuous and exponentially bounded evolution family on X. Let J be a nonnegative functional on the positive cone of the space of all realvalued locally bounded functions on R+: = [0,∞). We suppose that J satisfies some extraassumptions. Then the family U is uniformly exponentially stable provided that for every x ∈ X we have: sup s≥0 J(U(s+ ·, s)x) <∞. This result is connected to the uniform asymptotic stability of the wellposed linear and nonautonomous abstract Cauchy problem{ u̇(t) = A(t)u(t), t ≥ s ≥ 0, u(s) = x x ∈ X. In the autonomous case, i.e. when U(t, s) = T (t − s) for some strongly continuous semigroup {T (t)}t≥0 we obtain the wellknown theorems of Datko, Littman, Neerven, Pazy and Rolewicz. 1.
Fredholm properties of evolution semigroups
 ILLINOIS J. MATH
, 2004
"... We show that the Fredholm spectrum of an evolution semigroup {Et}t≥0 is equal to its spectrum, and prove that the ranges of the operator Et − I and the generator G of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banac ..."
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We show that the Fredholm spectrum of an evolution semigroup {Et}t≥0 is equal to its spectrum, and prove that the ranges of the operator Et − I and the generator G of the evolution semigroup are closed simultaneously. The evolution semigroup is acting on spaces of functions with values in a Banach space, and is induced by an evolution family that could be the propagator for a wellposed linear differential equation u′(t) = A(t)u(t) with, generally, unbounded operators A(t); in this case G is the closure of the operator G given by (Gu)(t) = −u′(t) +A(t)u(t).
WELLPOSEDNESS OF HYPERBOLIC EVOLUTION EQUATIONS IN BANACH SPACES
, 2005
"... Abstract. We study wellposedness of hyperbolic nonautonomous linear evolution equations u ′ (t) = A(t)u(t) in Banach spaces X. Using the theory of evolution semigroups, we develop two different notions of solvability, examine their properties and give existence and uniqueness theorems. At first we ..."
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Abstract. We study wellposedness of hyperbolic nonautonomous linear evolution equations u ′ (t) = A(t)u(t) in Banach spaces X. Using the theory of evolution semigroups, we develop two different notions of solvability, examine their properties and give existence and uniqueness theorems. At first we consider solutions which are limits of classical solutions. This concept is seen to coincide with weak solutions in our setting. Second, we study limits of solutions to suitable approximating problems. Here we obtain for separable Hilbert spaces X and skew adjoint operators A(t) an existence and uniqueness result under minimal additional assumptions. We apply our results to examples motivated from quantum theory. In particular, we show the existence of the time evolution in the theory of a massive bosonic quantum field with localized polynomial selfinteraction on two dimensional space time. 1.
Two classes of passive timevarying wellposed linear systems
"... Abstract. We investigate two classes of timevarying wellposed linear systems. Starting from a timeinvariant scatteringpassive system, each of the timevarying systems is constructed by introducing a timedependent inner product on the state space and modifying some of the generating operators. ..."
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Abstract. We investigate two classes of timevarying wellposed linear systems. Starting from a timeinvariant scatteringpassive system, each of the timevarying systems is constructed by introducing a timedependent inner product on the state space and modifying some of the generating operators. These classes of linear systems are motivated by physical examples such as the electromagnetic field around a moving object. To prove the wellposedness of these systems, we use the LaxPhillips semigroup induced by a wellposed linear system, as in scattering theory. Key words. Wellposed linear system, operator semigroup, linear timevarying system, LaxPhillips semigroup, scattering passive system.
L p –REGULARITY FOR PARABOLIC OPERATORS WITH UNBOUNDED TIME–DEPENDENT COEFFICIENTS
, 903
"... Abstract. We establish the maximal regularity for nonautonomous OrnsteinUhlenbeck operators in L pspaces with respect to a family of invariant measures, where p ∈ (1, +∞). This result follows from the maximal L pregularity for a class of elliptic operators with unbounded, timedependent drift coe ..."
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Abstract. We establish the maximal regularity for nonautonomous OrnsteinUhlenbeck operators in L pspaces with respect to a family of invariant measures, where p ∈ (1, +∞). This result follows from the maximal L pregularity for a class of elliptic operators with unbounded, timedependent drift coefficients and potentials acting on L p (R N) with Lebesgue measure. 1.
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"... A spectral mapping theorem for evolution semigroups on asymptotically almost periodic functions defined on the half line ∗ ..."
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A spectral mapping theorem for evolution semigroups on asymptotically almost periodic functions defined on the half line ∗
ftp ejde.math.txstate.edu (login: ftp) NEW CHARACTERIZATIONS OF ASYMPTOTIC STABILITY FOR EVOLUTION FAMILIES ON BANACH SPACES
"... Abstract. We generalize the Datko Rolewicz theorem on exponential stability in the nonautonomous case. Also, we extend the results obtained by Jan van Neerven [18]. 1. ..."
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Abstract. We generalize the Datko Rolewicz theorem on exponential stability in the nonautonomous case. Also, we extend the results obtained by Jan van Neerven [18]. 1.