If the norm of the time-dependent unbounded perturbation operators is continuous, the nonautonomous evolution equation admits an evolution family, unique under additional differentiability and bounded derivative conditions.
Megdiche, A new generation criterion theorem for C 0-semigroups implying a generalization of Kaiser-Weis-Batty’s perturbation theorem,Semigroup Forum107(2023), 491–524
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On the well-posedness of linear evolution equations under unbounded nonautonomous perturbations
If the norm of the time-dependent unbounded perturbation operators is continuous, the nonautonomous evolution equation admits an evolution family, unique under additional differentiability and bounded derivative conditions.