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.
Rau, Hyperbolic evolutionary semigroups on vector valued function spaces,Semigroup Forum 48(1994), 107–118
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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.