Maximal Lp-regularity for x-dependent fractional heat equations with Dirichlet conditions

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 P of order 2a (0 < a < 1) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian (-Delta)(a), as well as elliptic operators (-del & sdot; A(x) del + b(x))(a). The proofs are based on general results on maximal L-p-regularity and its relation to 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 in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.