$L^p(I,C^\alpha(\Omega))$ Regularity for Reaction-Diffusion Equations with Non-smooth Data

We prove an $L^p(I,C^\alpha(\Omega))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^\alpha(\Omega))$ norm with respect to the data. To prove our result, we first establish $C^\alpha(\Omega)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).