We consider the maximal regularity problem for non-autonomous parabolic equations \(\begin{aligned} u'(t) + \mathcal {A}(t) u(t) = f(t)\ \, \, t\text {-a.e.}, \ u(0) = u_0. \end{aligned}\) Each operator \( \mathcal {A}(t)\) arises from a time depending sesquilinear form \(\mathfrak {a}(t)\) on a Hilbert space \(\mathcal {H}\) with constant domain \(\mathcal {V}\) . We prove maximal \(L^p\) -regularity result for \(1<p\le 2\) under minimal regularity assumptions on the forms. Our main assumption is that \((\mathcal {A}(t))_{t\in [0,\tau ]}\) are piecewise in the Besov space \(B^{\frac{1}{2},2}_p\) with respect to the variable t. This improves previously known results. We give four examples that illustrate our results.
