Stability theory for semigroups using $(L^{p},L^{q})$ Fourier multipliers

TL;DR

This paper develops a new approach to analyze the stability of $C_{0}$-semigroups using $(L^{p},L^{q})$ Fourier multipliers, providing novel decay rate characterizations and unifying existing stability results.

Contribution

It introduces a characterization of polynomial decay of semigroup orbits via $(L^{p},L^{q})$ Fourier multipliers, extending stability analysis without assuming uniform boundedness.

Findings

Derived new polynomial decay rates depending on space geometry

Reproved and unified existing exponential stability results

Established a new exponential stability theorem for positive semigroups

Abstract

We study polynomial and exponential stability for $C_{0}$-semigroups using the recently developed theory of operator-valued $(L^{p},L^{q})$ Fourier multipliers. We characterize polynomial decay of orbits of a $C_{0}$-semigroup in terms of the $(L^{p},L^{q})$ Fourier multiplier properties of its resolvent. Using this characterization we derive new polynomial decay rates which depend on the geometry of the underlying space. We do not assume that the semigroup is uniformly bounded, our results depend only on spectral properties of the generator. As a corollary of our work on polynomial stability we reprove and unify various existing results on exponential stability, and we also obtain a new theorem on exponential stability for positive semigroups.