Fractional powers approach of operators for higher order abstract Cauchy problems

TL;DR

This paper investigates fractional powers of non-negative operators and their connection to Chebyshev polynomials to analyze existence, regularity, and asymptotic behavior of solutions for higher-order abstract Cauchy problems.

Contribution

It introduces a novel approach linking fractional operator powers with Chebyshev polynomials to study higher-order evolution equations.

Findings

Established existence and regularity results for solutions of higher-order abstract Cauchy problems.

Generalized classical results on structural damping in linear differential systems.

Provided asymptotic behavior analysis for solutions of these equations.

Abstract

In this paper we explore the theory of fractional powers of non-negative (and not necessarily self-adjoint) operators and its amazing relationship with the Chebyshev polynomials of the second kind to obtain results of existence, regularity and behavior asymptotic of solutions for linear abstract evolution equations of $n$-th order in time, where $n\geqslant3$. We also prove generalizations of classical results on structural damping for linear systems of differential equations.