Maximal \(L_p\)-regularity for x-dependent fractional heat equations with Dirichlet conditions

We prove optimal regularity results in \(L_p\) $L_p$ -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 ( \(0<a<1\) $0<a<1$ ) with nonsmooth x-dependent coefficients. This includes the prominent case of the fractional Laplacian \((-\Delta )^a\) ${(-\mathrm{\Delta })}^a$ , as well as elliptic operators \((-\nabla \cdot A(x)\nabla +b(x))^a\) ${(-\mathrm{\nabla }·A(x)\mathrm{\nabla }+b(x))}^a$ . The proofs are based on general results on maximal \(L_p\) $L_p$ -regularity and its relation to \(\mathcal {R}\) $\mathcal{R}$ -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 in the case of domains with boundary; the linear results are so when P is x-dependent nonsymmetric.