Approximation and asymptotic behaviour of evolution families

摘要

Let (A(t))_(t≥0) and (B(t))_(t≥0) we two families of closed operators satisfying the Acquistapace-Terreni conditions or the Kato-Tanabe conditions, or assumptions of maximal regularity, and let (U(t, s))_(t>s≥0) and (V(T, S))_(t>s≥0) be the associated evolution families. We obtain some estimates for ‖U(t, s) - V(t, s)‖ in terms of ‖A(τ)~(-1) - B(τ)~(-1)‖ for s ≤ τ ≤ t. We deduce some results showing that if ‖A(τ)~(-1) - B(τ)~(-1)‖ → 0 sufficiently quickly as τ → ∞ then U and V have similar asymptotic behaviour.