In this paper, we present counterexamples to maximal Lp-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 L2-regularity on Hβ1 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 Lp-regularity
on Hβ1(Rd) or L2-regularity on L2(Rd)