  We prove optimal regularity results in $L_p$-based function spaces in space
and time for a large class of linear parabolic equations with a nonlocal
elliptic operator in bounded domains with limited smoothness. Here the nonlocal
operator is given by a strongly elliptic and even pseudodifferential operator
of order $2a$ ($0<a<1$) with nonsmooth $x$-dependent coefficients. The proofs
are based on general results on maximal $L_p$-regularity and its relation to
$\mathcal{R}$-boundedness of the resolvent of the associated (elliptic)
operator. Finally, we apply these results to show existence of strong solutions
locally in time for a class of nonlinear nonlocal parabolic equations, which
include a fractional nonlinear diffusion equation and a fractional porous
medium equation after a transformation.
  The nonlinear results are new for operators on domains with boundary; the
linear results are so when $P$ is $x$-dependent nonsymmetric.
