Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations $$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$ u ′ ( t ) + A ( t ) u ( t ) = F ( t , u ) , t -a.e. , u ( 0 ) = u 0 . Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $$\mathcal {H}.$$ H . We prove the maximal regularity result in temporally weighted $$L^2$$ L 2 -spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value $$u_0$$ u 0 and the inhomogeneous term F. Our results are motivated by boundary value problems.