Advances in Differential Equations

Linear non-autonomous Cauchy problems and evolution semigroups

Hagen Neidhardt and Valentin A. Zagrebnov

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The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

Article information

Adv. Differential Equations, Volume 14, Number 3/4 (2009), 289-340.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35L90: Abstract hyperbolic equations 34G10: Linear equations [See also 47D06, 47D09] 47D06: One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]


Neidhardt, Hagen; Zagrebnov, Valentin A. Linear non-autonomous Cauchy problems and evolution semigroups. Adv. Differential Equations 14 (2009), no. 3/4, 289--340.

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