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

Abstract

We consider the problem of maximal regularity for the semilinear non-autonomous evolution equations

$$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$

Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space \(\mathcal {H}.\) We prove the maximal regularity result in temporally weighted \(L^2\)-spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value \(u_0\) and the inhomogeneous term F. Our results are motivated by boundary value problems.