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-Voigt-type model for visco-elastic solids.

On the well‐posedness of a class of nonautonomous SPDEs: An operator‐theoretical perspective

“…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 .…”

“…[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.…”

“…We can apply Picard et al [13,Lemma 2.6] to M 0 (m) = (1 − P), M 1 (m) = P, A = 0 to obtain for all t ∈ R, > 0: Since ,  ′ ,  are diagonal block operator matrices, it suffices to restrict ourselves to test functions inC ∞ (R; H 1 ) andC ∞ (R; H 2 ).…”

Some remarks on the notions of boundary systems and boundary triple(t)s

“…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 and that a skew‐self‐adjoint restriction of automatically satisfies .…”

On abstract grad–div systems

“…[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].…”