Maximal $L_p$-regularity for fractional problem driven by non-autonomous forms

We investigate the maximal $L_p$-regularity in J.L. Lions’ problem involving a time-fractional derivative and a non-autonomous form $a(t;\cdot ,\cdot )$ on a Hilbert space $H$. This problem says whether the maximal $L_p$-regularity in $H$ hold when $t\mapsto a(t;u,v)$ is merely continuous or even merely measurable. We prove the maximal $L_p$-regularity results when the coefficients satisfy general Dini-type continuity conditions. In particular, we construct a counterexample to negatively answer this problem, indicating the minimal Hölder-scale regularity required for positive results.