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

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.

### 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
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