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.
Henry,Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics840, Springer-Verlag, Berlin-New York
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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.