On Non-Autonomous Evolutionary Problems

TL;DR

This paper extends well-posedness results for evolutionary problems to non-autonomous media, providing an elementary Hilbert space approach applicable to a broad class of equations, including visco-elastic models.

Contribution

It introduces a new method for analyzing non-autonomous evolutionary equations using Hilbert spaces, overcoming limitations of traditional evolution family strategies.

Findings

Successfully extended well-posedness to non-autonomous problems

Applied approach to Kelvin-Voigt visco-elastic model

Provided a unified framework for diverse evolutionary equations

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.