Direct and inverse abstract Cauchy problems with fractional powers of almost sectorial operators

TL;DR

This paper develops explicit solution operators for abstract Cauchy problems involving fractional powers of almost sectorial operators with time-variable coefficients, enabling solutions to inverse problems and time-dependent operator cases.

Contribution

It introduces explicit solution operators for fractional power Cauchy problems with time-variable coefficients and extends to time-dependent almost sectorial operators.

Findings

Explicit solution operator derived for fractional power Cauchy problems

Inverse problem solution via operator representation

Extension to time-dependent almost sectorial operators

Abstract

We derive the explicit solution operator of an abstract Cauchy problem involving a time-variable coefficient and a fractional power of an almost sectorial operator. The time-variable coefficient is recovered by solving the inverse abstract Cauchy problem using the solution operator representation. As a complement, we also study similar problems by considering almost sectorial operators that depend on a time-variable.