  Optimal second-order regularity in the space variables is established for
solutions to Cauchy-Dirichlet problems for nonlinear parabolic equations and
systems of $p$-Laplacian type, with square-integrable right-hand sides and
initial data in a Sobolev space. As a consequence, generalized solutions are
shown to be strong solutions. Minimal regularity on the boundary of the domain
is required, though the results are new even for smooth domains. In particular,
they hold in arbitrary bounded convex domains.
