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
