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.
Widder,The Laplace Transform(2nd Ed.), Princeton University Press, Princeton
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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.