Linear non-autonomous Cauchy problems and evolution semigroups. (English) Zbl 1180.35371

Neidhardt, Hagen; Zagrebnov, Valentin A.

Adv. Differ. Equ. 14, No. 3-4, 289-340 (2009).

The paper is devoted to the Cauchy problem for the non-autonomous evolution equation
\[ {{\partial u(t)}\over{\partial t}}+A(t)u(t)=0, \quad u(s)=u_s\in X, \quad t, s \in{\mathcal I}. \tag{1} \]
of the hyperbolic type in a separable Banach space \(X\). In the introduction a good survey on different approaches to construction solution operators or propagators \(U(t,s):\;u(t):=U(t,s)u_0\) for non-autonomous equations is given. Special attention is paid to standard methods based on different approximations of (1) and to the extension method based on an extension of an evolution operator \(\widetilde{K}_{\mathcal I}\) to the generator of a semigroup \({\mathcal U}(\sigma), \sigma\geq0,\) in \(L^p({\mathcal I}, X)\) related to the propagators:
\[ {\mathcal U}(\sigma)f(t):=U(t-\sigma)\chi_{\mathcal I}(t-\sigma)f(t-\sigma), \quad f\in L^p({\mathcal I}, X), \sigma\geq0. \]
The second approach is under investigation and underlies main results of the authors. It is proved that if the evolution operator \(\widetilde{K}_{\mathcal I}\) is closable in \(L^p({\mathcal I}, X)\) and its closure is an unti-generator, then (1) has an unique solution; conditions for closedness of \(\widetilde{K}_{\mathcal I}\) are given.
Obtained results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.

Reviewer: Irina V. Melnikova (Ekaterinburg)

MSC:

35L90	Abstract hyperbolic equations
34G10	Linear differential equations in abstract spaces
47D06	One-parameter semigroups and linear evolution equations

Keywords:

propagator; generator; extension method