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An old problem due to J.-L. Lions going back to the 1960s asks whether theabstract Cauchy problem associated to non-autonomous forms has maximalregularity if the time dependence is merely assumed to be continuous or evenmeasurable. We give a negative answer to this question and discuss the minimalregularity needed for positive results.

J.-L. Lions' Problem Concerning Maximal Regularity of Equations Governed by Non-Autonomous Forms