2013
DOI: 10.1007/s00028-013-0201-7
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On non-autonomous evolutionary problems

Abstract: The paper extends well-posedness results of a previously explored class of time-shift invariant evolutionary problems to the case of non-autonomous media. The Hilbert space setting developed for the time-shift invariant case can be utilized to obtain an elementary approach to non-autonomous equations. The results cover a large class of evolutionary equations, where well-known strategies like evolution families may be difficult to use or fail to work. We exemplify the approach with an application to a Kelvin-Vo… Show more

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Cited by 31 publications
(89 citation statements)
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References 20 publications
(21 reference statements)
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“…Moreover, we note that ‖b ′ (m)‖ ≤ ‖b‖ Lip , by Picard et al [13,Lemma 2.1] Hence, ‖b ′ (m)‖ ≤ 1 c 2 ‖a‖ Lip . Moreover, we note that ‖b ′ (m)‖ ≤ ‖b‖ Lip , by Picard et al [13,Lemma 2.1] Hence, ‖b ′ (m)‖ ≤ 1 c 2 ‖a‖ Lip .…”
Section: An Abstract Stochastic Heat/wave Equationmentioning
confidence: 67%
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“…Moreover, we note that ‖b ′ (m)‖ ≤ ‖b‖ Lip , by Picard et al [13,Lemma 2.1] Hence, ‖b ′ (m)‖ ≤ 1 c 2 ‖a‖ Lip . Moreover, we note that ‖b ′ (m)‖ ≤ ‖b‖ Lip , by Picard et al [13,Lemma 2.1] Hence, ‖b ′ (m)‖ ≤ 1 c 2 ‖a‖ Lip .…”
Section: An Abstract Stochastic Heat/wave Equationmentioning
confidence: 67%
“…[11] Later on, this has been generalized to nonautonomous or nonlinear equations, see for example Trostorff et al [7,[12][13][14] To keep this article conveniently self-contained, we shall summarize the well-posedness theorem outlined in Waurick. [11] Later on, this has been generalized to nonautonomous or nonlinear equations, see for example Trostorff et al [7,[12][13][14] To keep this article conveniently self-contained, we shall summarize the well-posedness theorem outlined in Waurick.…”
Section: The Deterministic Solution Theorymentioning
confidence: 99%
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“…Picard et al. [, footnote on p. 751], that in the literature the term skew‐adjoint seems to be more common although this “binary expression” might cause confusion. Notice that a skew‐self‐adjoint extension H of H 0 automatically satisfies HH0 and that a skew‐self‐adjoint restriction of H0 automatically satisfies H0H.…”
Section: Skew‐self‐adjoint Extensions Of Skew‐symmetric Operatorsmentioning
confidence: 99%
“…[15,16]. The operator M is referred to as the material law operator, which in the situation discussed here is a linear operator acting on a Hilbert space realizing the space-time the problems are formulated in, [24,34,21,15].…”
Section: Introductionmentioning
confidence: 99%