Improved polynomial decay for unbounded semigroups

TL;DR

This paper establishes polynomial decay rates for unbounded $C_0$-semigroups based on polynomial resolvent growth, improving previous results by removing logarithmic losses on non-Hilbertian Banach spaces.

Contribution

It provides new decay estimates for unbounded semigroups without the need for uniform boundedness, enhancing prior results especially on Banach spaces.

Findings

Polynomial decay rates are achieved under polynomial resolvent growth.

Previous logarithmic losses are eliminated for non-Hilbertian Banach spaces.

Results apply to unbounded $C_0$-semigroups without uniform boundedness.

Abstract

We obtain polynomial decay rates for $C_{0}$-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces.