Solvability of initial-boundary value problems for non-autonomous evolution equations

TL;DR

This paper investigates the solvability of initial-boundary value problems for non-autonomous evolution equations, establishing existence, uniqueness, and regularity under minimal continuity assumptions on time-dependent operators.

Contribution

It provides a unified framework for analyzing linear non-autonomous evolution equations with time-varying domains, using only continuity assumptions for the main results.

Findings

Proved existence and uniqueness of solutions.

Established maximal regularity in Sobolev spaces.

Applicable to equations with changing operator domains.

Abstract

The initial-boundary value problems for linear non-autonomous first order evolution equations are examined. Our assumptions provide a unified treatment which is applicable to many situations, where the domains of the operators may change with time. We study existence, uniqueness and maximal regularity of solutions in Sobolev spaces. In contrast to the previous results we use only the continuity assumption on the operators in the main part of the equation.