Stability of Semigroups of Linear Operators in Variable Banach Spaces

Abstract

This paper develops a theory for stability analysis of semigroups of linear operators acting on variable Banach spaces—families of Banach spaces {X(t)}t≥0 whose norms may depend on time. We establish generation theorems under appropriate resolvent conditions, characterize exponential stability through Lyapunov-type functionals, and analyze spectral properties of evolution families in variable settings. Our approach systematically extends classical semigroup theory to accommodate time-dependent norms by transporting all objects to a fixed reference space. Applications include non-autonomous parabolic equations and reaction-diffusion systems with time-dependent coefficients. All proofs are provided with full mathematical rigor, addressing technical challenges unique to variable Banach spaces