  In this paper, we present counterexamples to maximal $L^p$-regularity for a
parabolic PDE. The example is a second-order operator in divergence form with
space and time-dependent coefficients. It is well-known from Lions' theory that
such operators admit maximal $L^2$-regularity on $H^{-1}$ under a coercivity
condition on the coefficients, and without any regularity conditions in time
and space. We show that in general one cannot expect maximal $L^p$-regularity
on $H^{-1}(\mathbb{R}^d)$ or $L^2$-regularity on $L^2(\mathbb{R}^d)$.
